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AE4M33RZN Fuzzy logicTutorial examples
Radomiacuter Černochradomircernochfelcvutcz
Faculty of Electrical Engineering CTU in Prague
2012
Task 2
AssignmentOn the universeΔ = a b c d there is a fuzzy set
120583A = 1114106(a 03) (b 1) (c 05)1114109
Find its horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
X 120572 = 0a b c 120572 isin (0 03⟩b c 120572 isin (03 05⟩b 120572 isin (05 1⟩
Task 3
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩
Find the vertical representation
Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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AE4M33RZN Fuzzy logicTutorial examples
Radomiacuter Černochradomircernochfelcvutcz
Faculty of Electrical Engineering CTU in Prague
2012
Task 2
AssignmentOn the universeΔ = a b c d there is a fuzzy set
120583A = 1114106(a 03) (b 1) (c 05)1114109
Find its horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
X 120572 = 0a b c 120572 isin (0 03⟩b c 120572 isin (03 05⟩b 120572 isin (05 1⟩
Task 3
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩
Find the vertical representation
Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 2
AssignmentOn the universeΔ = a b c d there is a fuzzy set
120583A = 1114106(a 03) (b 1) (c 05)1114109
Find its horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
X 120572 = 0a b c 120572 isin (0 03⟩b c 120572 isin (03 05⟩b 120572 isin (05 1⟩
Task 3
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩
Find the vertical representation
Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 4: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/4.jpg)
Task 3
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩
Find the vertical representation
Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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![Page 6: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/6.jpg)
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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![Page 7: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/7.jpg)
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 8: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/8.jpg)
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 9: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/9.jpg)
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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![Page 12: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/12.jpg)
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 13: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/13.jpg)
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 14: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/14.jpg)
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 15: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/15.jpg)
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 16: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/16.jpg)
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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![Page 18: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/18.jpg)
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 19: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/19.jpg)
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 20: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/20.jpg)
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 21: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/21.jpg)
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
![Page 22: OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague](https://reader036.vdocument.in/reader036/viewer/2022081407/5f8220c366aac100580433b9/html5/thumbnails/22.jpg)
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