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Discrete MathematicsAlgebraic Structures
H. Turgut Uyar Aysegul Gencata Yayımlı Emre Harmancı
2001-2012
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License
c©2001-2012 T. Uyar, A. Yayımlı, E. Harmancı
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Topics
1 Algebraic StructuresIntroductionGroupsRings
2 LatticesPartially Ordered SetsLatticesBoolean Algebra
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Topics
1 Algebraic StructuresIntroductionGroupsRings
2 LatticesPartially Ordered SetsLatticesBoolean Algebra
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Algebraic Structure
algebraic structure: <set, operations, constants>
carrier setoperations: binary, unaryconstants: identity, zero
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Operation
every operation is a function
binary operation:◦ : S × S → T
unary operation:∆ : S → T
closed: T ⊆ S
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Operation
every operation is a function
binary operation:◦ : S × S → T
unary operation:∆ : S → T
closed: T ⊆ S
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Closed Operation Examples
Example
subtraction is closed on Zsubtraction is not closed on Z+
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Closed Operation Examples
Example
subtraction is closed on Zsubtraction is not closed on Z+
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Binary Operation Properties
Definition
commutativity:∀a, b ∈ S a ◦ b = b ◦ a
Definition
associativity:∀a, b, c ∈ S (a ◦ b) ◦ c = a ◦ (b ◦ c)
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Binary Operation Example
Example
◦ : Z× Z → Za ◦ b = a + b − 3ab
commutative:a ◦ b = a + b − 3ab = b + a− 3ba = b ◦ a
associative:(a ◦ b) ◦ c = (a + b − 3ab) + c − 3(a + b − 3ab)c
= a + b − 3ab + c − 3ac − 3bc + 9abc= a + b + c − 3ab − 3ac − 3bc + 9abc= a + (b + c − 3bc)− 3a(b + c − 3bc)= a ◦ (b ◦ c)
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Binary Operation Example
Example
◦ : Z× Z → Za ◦ b = a + b − 3ab
commutative:a ◦ b = a + b − 3ab = b + a− 3ba = b ◦ a
associative:(a ◦ b) ◦ c = (a + b − 3ab) + c − 3(a + b − 3ab)c
= a + b − 3ab + c − 3ac − 3bc + 9abc= a + b + c − 3ab − 3ac − 3bc + 9abc= a + (b + c − 3bc)− 3a(b + c − 3bc)= a ◦ (b ◦ c)
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Binary Operation Example
Example
◦ : Z× Z → Za ◦ b = a + b − 3ab
commutative:a ◦ b = a + b − 3ab = b + a− 3ba = b ◦ a
associative:(a ◦ b) ◦ c = (a + b − 3ab) + c − 3(a + b − 3ab)c
= a + b − 3ab + c − 3ac − 3bc + 9abc= a + b + c − 3ab − 3ac − 3bc + 9abc= a + (b + c − 3bc)− 3a(b + c − 3bc)= a ◦ (b ◦ c)
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Binary Operation Example
Example
◦ : Z× Z → Za ◦ b = a + b − 3ab
commutative:a ◦ b = a + b − 3ab = b + a− 3ba = b ◦ a
associative:(a ◦ b) ◦ c = (a + b − 3ab) + c − 3(a + b − 3ab)c
= a + b − 3ab + c − 3ac − 3bc + 9abc= a + b + c − 3ab − 3ac − 3bc + 9abc= a + (b + c − 3bc)− 3a(b + c − 3bc)= a ◦ (b ◦ c)
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Binary Operation Example
Example
◦ : Z× Z → Za ◦ b = a + b − 3ab
commutative:a ◦ b = a + b − 3ab = b + a− 3ba = b ◦ a
associative:(a ◦ b) ◦ c = (a + b − 3ab) + c − 3(a + b − 3ab)c
= a + b − 3ab + c − 3ac − 3bc + 9abc= a + b + c − 3ab − 3ac − 3bc + 9abc= a + (b + c − 3bc)− 3a(b + c − 3bc)= a ◦ (b ◦ c)
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Binary Operation Example
Example
◦ : Z× Z → Za ◦ b = a + b − 3ab
commutative:a ◦ b = a + b − 3ab = b + a− 3ba = b ◦ a
associative:(a ◦ b) ◦ c = (a + b − 3ab) + c − 3(a + b − 3ab)c
= a + b − 3ab + c − 3ac − 3bc + 9abc= a + b + c − 3ab − 3ac − 3bc + 9abc= a + (b + c − 3bc)− 3a(b + c − 3bc)= a ◦ (b ◦ c)
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Binary Operation Example
Example
◦ : Z× Z → Za ◦ b = a + b − 3ab
commutative:a ◦ b = a + b − 3ab = b + a− 3ba = b ◦ a
associative:(a ◦ b) ◦ c = (a + b − 3ab) + c − 3(a + b − 3ab)c
= a + b − 3ab + c − 3ac − 3bc + 9abc= a + b + c − 3ab − 3ac − 3bc + 9abc= a + (b + c − 3bc)− 3a(b + c − 3bc)= a ◦ (b ◦ c)
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Binary Operation Example
Example
◦ : Z× Z → Za ◦ b = a + b − 3ab
commutative:a ◦ b = a + b − 3ab = b + a− 3ba = b ◦ a
associative:(a ◦ b) ◦ c = (a + b − 3ab) + c − 3(a + b − 3ab)c
= a + b − 3ab + c − 3ac − 3bc + 9abc= a + b + c − 3ab − 3ac − 3bc + 9abc= a + (b + c − 3bc)− 3a(b + c − 3bc)= a ◦ (b ◦ c)
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Binary Operation Example
Example
◦ : Z× Z → Za ◦ b = a + b − 3ab
commutative:a ◦ b = a + b − 3ab = b + a− 3ba = b ◦ a
associative:(a ◦ b) ◦ c = (a + b − 3ab) + c − 3(a + b − 3ab)c
= a + b − 3ab + c − 3ac − 3bc + 9abc= a + b + c − 3ab − 3ac − 3bc + 9abc= a + (b + c − 3bc)− 3a(b + c − 3bc)= a ◦ (b ◦ c)
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Constants
Definition
identity:x ◦ 1 = 1 ◦ x = x
left identity: 1l ◦ x = x
right identity: x ◦ 1r = x
Definition
zero:x ◦ 0 = 0 ◦ x = 0
left zero: 0l ◦ x = 0
right zero: x ◦ 0r = 0
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Constants
Definition
identity:x ◦ 1 = 1 ◦ x = x
left identity: 1l ◦ x = x
right identity: x ◦ 1r = x
Definition
zero:x ◦ 0 = 0 ◦ x = 0
left zero: 0l ◦ x = 0
right zero: x ◦ 0r = 0
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Examples of Constants
Example
identity for < N,max > is 0
zero for < N,min > is 0
zero for < Z+,min > is 1
Example
◦ a b c
a a b b
b a b c
c a b a
b is a left identity
a and b are right zeros
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Examples of Constants
Example
identity for < N,max > is 0
zero for < N,min > is 0
zero for < Z+,min > is 1
Example
◦ a b c
a a b b
b a b c
c a b a
b is a left identity
a and b are right zeros
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Constants
Theorem
∃1l ∧ ∃1r ⇒ 1l = 1r
Proof.
1l ◦ 1r = 1l = 1r
Theorem
∃0l ∧ ∃0r ⇒ 0l = 0r
Proof.
0l ◦ 0r = 0l = 0r
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Constants
Theorem
∃1l ∧ ∃1r ⇒ 1l = 1r
Proof.
1l ◦ 1r = 1l = 1r
Theorem
∃0l ∧ ∃0r ⇒ 0l = 0r
Proof.
0l ◦ 0r = 0l = 0r
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Constants
Theorem
∃1l ∧ ∃1r ⇒ 1l = 1r
Proof.
1l ◦ 1r = 1l = 1r
Theorem
∃0l ∧ ∃0r ⇒ 0l = 0r
Proof.
0l ◦ 0r = 0l = 0r
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Constants
Theorem
∃1l ∧ ∃1r ⇒ 1l = 1r
Proof.
1l ◦ 1r = 1l = 1r
Theorem
∃0l ∧ ∃0r ⇒ 0l = 0r
Proof.
0l ◦ 0r = 0l = 0r
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Constants
Theorem
∃1l ∧ ∃1r ⇒ 1l = 1r
Proof.
1l ◦ 1r = 1l = 1r
Theorem
∃0l ∧ ∃0r ⇒ 0l = 0r
Proof.
0l ◦ 0r = 0l = 0r
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Constants
Theorem
∃1l ∧ ∃1r ⇒ 1l = 1r
Proof.
1l ◦ 1r = 1l = 1r
Theorem
∃0l ∧ ∃0r ⇒ 0l = 0r
Proof.
0l ◦ 0r = 0l = 0r
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Constants
Theorem
∃1l ∧ ∃1r ⇒ 1l = 1r
Proof.
1l ◦ 1r = 1l = 1r
Theorem
∃0l ∧ ∃0r ⇒ 0l = 0r
Proof.
0l ◦ 0r = 0l = 0r
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Constants
Theorem
∃1l ∧ ∃1r ⇒ 1l = 1r
Proof.
1l ◦ 1r = 1l = 1r
Theorem
∃0l ∧ ∃0r ⇒ 0l = 0r
Proof.
0l ◦ 0r = 0l = 0r
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Inverse
if x ◦ y = 1:
x is a left inverse of yy is a right inverse of x
if x ◦ y = y ◦ x = 1 x and y are inverse
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Inverse
Theorem
if the operation ◦ is associative:w ◦ x = x ◦ y = 1 ⇒ w = y
Proof.w = w ◦ 1
= w ◦ (x ◦ y)= (w ◦ x) ◦ y= 1 ◦ y= y
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Inverse
Theorem
if the operation ◦ is associative:w ◦ x = x ◦ y = 1 ⇒ w = y
Proof.w = w ◦ 1
= w ◦ (x ◦ y)= (w ◦ x) ◦ y= 1 ◦ y= y
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Inverse
Theorem
if the operation ◦ is associative:w ◦ x = x ◦ y = 1 ⇒ w = y
Proof.w = w ◦ 1
= w ◦ (x ◦ y)= (w ◦ x) ◦ y= 1 ◦ y= y
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Inverse
Theorem
if the operation ◦ is associative:w ◦ x = x ◦ y = 1 ⇒ w = y
Proof.w = w ◦ 1
= w ◦ (x ◦ y)= (w ◦ x) ◦ y= 1 ◦ y= y
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Inverse
Theorem
if the operation ◦ is associative:w ◦ x = x ◦ y = 1 ⇒ w = y
Proof.w = w ◦ 1
= w ◦ (x ◦ y)= (w ◦ x) ◦ y= 1 ◦ y= y
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Inverse
Theorem
if the operation ◦ is associative:w ◦ x = x ◦ y = 1 ⇒ w = y
Proof.w = w ◦ 1
= w ◦ (x ◦ y)= (w ◦ x) ◦ y= 1 ◦ y= y
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Algebraic Families
algebraic family: algebraic structure, axioms
commutativity, associativityinverse elements
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Algebraic Family Examples
Example
axioms:
x ◦ y = y ◦ x(x ◦ y) ◦ z = x ◦ (y ◦ z)x ◦ 1 = x
structures for which these axioms hold:
< Z,+, 0 >< Z, ·, 1 >< P(S),∪, ∅ >
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Algebraic Family Examples
Example
axioms:
x ◦ y = y ◦ x(x ◦ y) ◦ z = x ◦ (y ◦ z)x ◦ 1 = x
structures for which these axioms hold:
< Z,+, 0 >< Z, ·, 1 >< P(S),∪, ∅ >
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Subalgebra
Definition
subalgebra:let A =< S , ◦,∆, k > ∧ A′ =< S ′, ◦′,∆′, k ′ >
A′ is a subalgebra of A if:
S ′ ⊆ S∀a, b ∈ S ′ a ◦′ b = a ◦ b ∈ S ′
∀a ∈ S ′ ∆′a = ∆a ∈ S ′
k ′ = k
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Subalgebra
Definition
subalgebra:let A =< S , ◦,∆, k > ∧ A′ =< S ′, ◦′,∆′, k ′ >
A′ is a subalgebra of A if:
S ′ ⊆ S∀a, b ∈ S ′ a ◦′ b = a ◦ b ∈ S ′
∀a ∈ S ′ ∆′a = ∆a ∈ S ′
k ′ = k
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Subalgebra Examples
Example
< Z+,+, 0 > is a subalgebra of < Z,+, 0 >.
< N,−, 0 > is not a subalgebra of < Z,−, 0 >.
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Subalgebra Examples
Example
< Z+,+, 0 > is a subalgebra of < Z,+, 0 >.
< N,−, 0 > is not a subalgebra of < Z,−, 0 >.
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Topics
1 Algebraic StructuresIntroductionGroupsRings
2 LatticesPartially Ordered SetsLatticesBoolean Algebra
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Semigroups
Definition
semigroup: < S , ◦ >
∀a, b, c ∈ S (a ◦ b) ◦ c = a ◦ (b ◦ c)
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Semigroup Examples
Example
< Σ+,& >
Σ: alphabet, Σ+: strings of length at least 1
&: string concatenation
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Monoids
Definition
monoid: < S , ◦, 1 >
∀a, b, c ∈ S (a ◦ b) ◦ c = a ◦ (b ◦ c)
∀a ∈ S a ◦ 1 = 1 ◦ a = a
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Monoid Examples
Example
< Σ∗,&, ε >
Σ: alphabet, Σ∗: strings of any length
&: string concatenation
ε: empty string
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Groups
Definition
group: < S , ◦, 1 >
∀a, b, c ∈ S (a ◦ b) ◦ c = a ◦ (b ◦ c)
∀a ∈ S a ◦ 1 = 1 ◦ a = a
∀a ∈ S ∃a−1 ∈ S a ◦ a−1 = a−1 ◦ a = 1
Abelian group: ∀a, b ∈ S a ◦ b = b ◦ a
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Groups
Definition
group: < S , ◦, 1 >
∀a, b, c ∈ S (a ◦ b) ◦ c = a ◦ (b ◦ c)
∀a ∈ S a ◦ 1 = 1 ◦ a = a
∀a ∈ S ∃a−1 ∈ S a ◦ a−1 = a−1 ◦ a = 1
Abelian group: ∀a, b ∈ S a ◦ b = b ◦ a
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Group Examples
Example
< Z,+, 0 > is a group.
< Q, ·, 1 > is not a group.
< Q− {0}, ·, 1 > is a group.
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Group Examples
Example
< Z,+, 0 > is a group.
< Q, ·, 1 > is not a group.
< Q− {0}, ·, 1 > is a group.
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Group Example: Permutation Composition
permutation: a bijective function on a set
representation: (a1 a2 . . . an
p(a1) p(a2) . . . p(an)
)
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Permutation Examples
Example
A = {1, 2, 3}
p1 =
(1 2 31 2 3
)p2 =
(1 2 31 3 2
)p3 =
(1 2 32 1 3
)p4 =
(1 2 32 3 1
)p5 =
(1 2 33 1 2
)p6 =
(1 2 33 2 1
)
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Group Example: Permutation Composition
permutation composition is associative
identity permutasyon: 1A(a1 a2 . . . an
a1 a2 . . . an
)
the set of permutations of the elements of a set,the permutation composition operationand the identity permutation constitute a group
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Group Example: Permutation Composition
permutation composition is associative
identity permutasyon: 1A(a1 a2 . . . an
a1 a2 . . . an
)
the set of permutations of the elements of a set,the permutation composition operationand the identity permutation constitute a group
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Group Example: Permutation Composition
Example (permutations on {1, 2, 3, 4})A 1A p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11
1 1 1 1 1 1 1 2 2 2 2 2 22 2 2 3 3 4 4 1 1 3 3 4 43 3 4 2 4 2 3 3 4 1 4 1 34 4 3 4 2 3 2 4 3 4 1 3 1
p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
1 3 3 3 3 3 3 4 4 4 4 4 42 1 1 2 2 4 4 1 1 2 2 3 33 2 4 1 4 1 2 2 3 1 3 1 24 4 2 4 1 2 1 3 2 3 1 2 1
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Group Example: Permutation Composition
Example
p8 � p12 = p12 � p8 = 1A:p12 = p−1
8 , p8 = p−112
p14 � p14 = 1A:p14 = p−1
14
G1 =< {1A, p1, . . . , p23}, �, 1A > is a group
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Group Example: Permutation Composition
Example
p8 � p12 = p12 � p8 = 1A:p12 = p−1
8 , p8 = p−112
p14 � p14 = 1A:p14 = p−1
14
G1 =< {1A, p1, . . . , p23}, �, 1A > is a group
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Group Example: Permutation Composition
Example
� 1A p2 p6 p8 p12 p14
1A 1A p2 p6 p8 p12 p14
p2 p2 1A p8 p6 p14 p12
p6 p6 p12 1A p14 p2 p8
p8 p8 p14 p2 p12 1A p6
p12 p12 p6 p14 1A p8 p2
p14 p14 p8 p12 p2 p6 1A
< {1A, p2, p6, p8, p12, p14}, �, 1A > is a subgroup of G1
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Left and Right Cancellation
Theorem
a ◦ c = b ◦ c ⇒ a = bc ◦ a = c ◦ b ⇒ a = b
Proof.a ◦ c = b ◦ c
⇒ (a ◦ c) ◦ c−1 = (b ◦ c) ◦ c−1
⇒ a ◦ (c ◦ c−1) = b ◦ (c ◦ c−1)⇒ a ◦ 1 = b ◦ 1⇒ a = b
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Left and Right Cancellation
Theorem
a ◦ c = b ◦ c ⇒ a = bc ◦ a = c ◦ b ⇒ a = b
Proof.a ◦ c = b ◦ c
⇒ (a ◦ c) ◦ c−1 = (b ◦ c) ◦ c−1
⇒ a ◦ (c ◦ c−1) = b ◦ (c ◦ c−1)⇒ a ◦ 1 = b ◦ 1⇒ a = b
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Left and Right Cancellation
Theorem
a ◦ c = b ◦ c ⇒ a = bc ◦ a = c ◦ b ⇒ a = b
Proof.a ◦ c = b ◦ c
⇒ (a ◦ c) ◦ c−1 = (b ◦ c) ◦ c−1
⇒ a ◦ (c ◦ c−1) = b ◦ (c ◦ c−1)⇒ a ◦ 1 = b ◦ 1⇒ a = b
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Left and Right Cancellation
Theorem
a ◦ c = b ◦ c ⇒ a = bc ◦ a = c ◦ b ⇒ a = b
Proof.a ◦ c = b ◦ c
⇒ (a ◦ c) ◦ c−1 = (b ◦ c) ◦ c−1
⇒ a ◦ (c ◦ c−1) = b ◦ (c ◦ c−1)⇒ a ◦ 1 = b ◦ 1⇒ a = b
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Left and Right Cancellation
Theorem
a ◦ c = b ◦ c ⇒ a = bc ◦ a = c ◦ b ⇒ a = b
Proof.a ◦ c = b ◦ c
⇒ (a ◦ c) ◦ c−1 = (b ◦ c) ◦ c−1
⇒ a ◦ (c ◦ c−1) = b ◦ (c ◦ c−1)⇒ a ◦ 1 = b ◦ 1⇒ a = b
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Left and Right Cancellation
Theorem
a ◦ c = b ◦ c ⇒ a = bc ◦ a = c ◦ b ⇒ a = b
Proof.a ◦ c = b ◦ c
⇒ (a ◦ c) ◦ c−1 = (b ◦ c) ◦ c−1
⇒ a ◦ (c ◦ c−1) = b ◦ (c ◦ c−1)⇒ a ◦ 1 = b ◦ 1⇒ a = b
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Basic Theorem of Groups
Theorem
The unique solution of the equation a ◦ x = b is: x = a−1 ◦ b.
Proof.a ◦ c = b
⇒ a−1 ◦ (a ◦ c) = a−1 ◦ b⇒ 1 ◦ c = a−1 ◦ b⇒ c = a−1 ◦ b
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Basic Theorem of Groups
Theorem
The unique solution of the equation a ◦ x = b is: x = a−1 ◦ b.
Proof.a ◦ c = b
⇒ a−1 ◦ (a ◦ c) = a−1 ◦ b⇒ 1 ◦ c = a−1 ◦ b⇒ c = a−1 ◦ b
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Basic Theorem of Groups
Theorem
The unique solution of the equation a ◦ x = b is: x = a−1 ◦ b.
Proof.a ◦ c = b
⇒ a−1 ◦ (a ◦ c) = a−1 ◦ b⇒ 1 ◦ c = a−1 ◦ b⇒ c = a−1 ◦ b
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Basic Theorem of Groups
Theorem
The unique solution of the equation a ◦ x = b is: x = a−1 ◦ b.
Proof.a ◦ c = b
⇒ a−1 ◦ (a ◦ c) = a−1 ◦ b⇒ 1 ◦ c = a−1 ◦ b⇒ c = a−1 ◦ b
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Basic Theorem of Groups
Theorem
The unique solution of the equation a ◦ x = b is: x = a−1 ◦ b.
Proof.a ◦ c = b
⇒ a−1 ◦ (a ◦ c) = a−1 ◦ b⇒ 1 ◦ c = a−1 ◦ b⇒ c = a−1 ◦ b
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Topics
1 Algebraic StructuresIntroductionGroupsRings
2 LatticesPartially Ordered SetsLatticesBoolean Algebra
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Ring
Definition
ring: < S ,+, ·, 0 >
∀a, b, c ∈ S (a + b) + c = a + (b + c)
∀a ∈ S a + 0 = 0 + a = a
∀a ∈ S ∃(−a) ∈ S a + (−a) = (−a) + a = 0
∀a, b ∈ S a + b = b + a
∀a, b, c ∈ S (a · b) · c = a · (b · c)
∀a, b, c ∈ S
a · (b + c) = a · b + a · c(b + c) · a = b · a + c · a
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Ring
Definition
ring: < S ,+, ·, 0 >
∀a, b, c ∈ S (a + b) + c = a + (b + c)
∀a ∈ S a + 0 = 0 + a = a
∀a ∈ S ∃(−a) ∈ S a + (−a) = (−a) + a = 0
∀a, b ∈ S a + b = b + a
∀a, b, c ∈ S (a · b) · c = a · (b · c)
∀a, b, c ∈ S
a · (b + c) = a · b + a · c(b + c) · a = b · a + c · a
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Ring
Definition
ring: < S ,+, ·, 0 >
∀a, b, c ∈ S (a + b) + c = a + (b + c)
∀a ∈ S a + 0 = 0 + a = a
∀a ∈ S ∃(−a) ∈ S a + (−a) = (−a) + a = 0
∀a, b ∈ S a + b = b + a
∀a, b, c ∈ S (a · b) · c = a · (b · c)
∀a, b, c ∈ S
a · (b + c) = a · b + a · c(b + c) · a = b · a + c · a
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Field
Definition
field: < S ,+, ·, 0, 1 >
all properties of a ring
∀a, b ∈ S a · b = b · a∀a ∈ S a · 1 = 1 · a = a
∀a ∈ S ∃a−1 ∈ S a · a−1 = a−1 · a = 1
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Field
Definition
field: < S ,+, ·, 0, 1 >
all properties of a ring
∀a, b ∈ S a · b = b · a∀a ∈ S a · 1 = 1 · a = a
∀a ∈ S ∃a−1 ∈ S a · a−1 = a−1 · a = 1
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References
Grimaldi
Chapter 5: Relations and Functions
5.4. Special Functions
Chapter 16: Groups, Coding Theory,and Polya’s Method of Enumeration
16.1. Definitions, Examples, and Elementary Properties
Chapter 14: Rings and Modular Arithmetic
14.1. The Ring Structure: Definition and Examples
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Topics
1 Algebraic StructuresIntroductionGroupsRings
2 LatticesPartially Ordered SetsLatticesBoolean Algebra
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Partially Ordered Set
Definition
partial order relation:
reflexive
anti-symmetric
transitive
partially ordered set (poset):a set with a partial order relation defined on its elements
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Partially Ordered Set
Definition
partial order relation:
reflexive
anti-symmetric
transitive
partially ordered set (poset):a set with a partial order relation defined on its elements
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Partial Order Examples
Example (set of sets, ⊆)
A ⊆ A
A ⊆ B ∧ B ⊆ A ⇒ A = B
A ⊆ B ∧ B ⊆ C ⇒ A ⊆ C
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Partial Order Examples
Example (Z, ≤)
x ≤ x
x ≤ y ∧ y ≤ x ⇒ x = y
x ≤ y ∧ y ≤ z ⇒ x ≤ z
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Partial Order Examples
Example (Z+, |)
x |xx |y ∧ y |x ⇒ x = y
x |y ∧ y |z ⇒ x |z
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Comparability
a � b: a precedes b
a � b ∨ b � a: a and b are comparable
total order (linear order):all elements are comparable with each other
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Comparability
a � b: a precedes b
a � b ∨ b � a: a and b are comparable
total order (linear order):all elements are comparable with each other
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Comparability Examples
Example
Z+, |: 3 and 5 are not comparable
Z,≤: total order
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Comparability Examples
Example
Z+, |: 3 and 5 are not comparable
Z,≤: total order
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Hasse Diagrams
a � b: a immediately precedes b¬∃x a � x � b
Hasse diagram:
draw a line between a and b if a � bpreceding element is below
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Hasse Diagrams
a � b: a immediately precedes b¬∃x a � x � b
Hasse diagram:
draw a line between a and b if a � bpreceding element is below
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Hasse Diagram Examples
Example
{1, 2, 3, 4, 6, 8, 9, 12, 18, 24}the relation |
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Consistent Enumeration
consistent enumeration:f : S → Na � b ⇒ f (a) ≤ f (b)
there can be more than one consistent enumeration
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Consistent Enumeration Examples
Example
{a 7−→ 5, b 7−→ 3, c 7−→ 4, d 7−→ 1, e 7−→ 2}{a 7−→ 5, b 7−→ 4, c 7−→ 3, d 7−→ 2, e 7−→ 1}
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Consistent Enumeration Examples
Example
{a 7−→ 5, b 7−→ 3, c 7−→ 4, d 7−→ 1, e 7−→ 2}{a 7−→ 5, b 7−→ 4, c 7−→ 3, d 7−→ 2, e 7−→ 1}
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Maximal - Minimal Elements
Definition
maximal element: max∀x ∈ S max � x ⇒ x = max
Definition
minimal element: min∀x ∈ S x � min ⇒ x = min
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Maximal - Minimal Elements
Definition
maximal element: max∀x ∈ S max � x ⇒ x = max
Definition
minimal element: min∀x ∈ S x � min ⇒ x = min
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Maximal - Minimal Element Examples
Example
max : 18, 24min : 1
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Bounds
Definition
A ⊆ S
M is an upper bound of A:∀x ∈ A x � M
M(A): set of upper bounds of A
sup(A) is the supremum of A:∀M ∈ M(A) sup(A) � M
Definition
A ⊆ S
m is a lower bound of A:∀x ∈ A m � x
m(A): set of lower bound of A
inf (A) is the infimum of A:∀m ∈ m(A) m � inf (A)
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Bounds
Definition
A ⊆ S
M is an upper bound of A:∀x ∈ A x � M
M(A): set of upper bounds of A
sup(A) is the supremum of A:∀M ∈ M(A) sup(A) � M
Definition
A ⊆ S
m is a lower bound of A:∀x ∈ A m � x
m(A): set of lower bound of A
inf (A) is the infimum of A:∀m ∈ m(A) m � inf (A)
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Bound Example
Example (factors of 36)
inf = greatest common divisorsup = least common multiple
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Topics
1 Algebraic StructuresIntroductionGroupsRings
2 LatticesPartially Ordered SetsLatticesBoolean Algebra
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Lattice
Definition
lattice: < L,∧,∨ >∧: meet, ∨: join
a ∧ b = b ∧ aa ∨ b = b ∨ a
(a ∧ b) ∧ c = a ∧ (b ∧ c)(a ∨ b) ∨ c = a ∨ (b ∨ c)
a ∧ (a ∨ b) = aa ∨ (a ∧ b) = a
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Lattice
Definition
lattice: < L,∧,∨ >∧: meet, ∨: join
a ∧ b = b ∧ aa ∨ b = b ∨ a
(a ∧ b) ∧ c = a ∧ (b ∧ c)(a ∨ b) ∨ c = a ∨ (b ∨ c)
a ∧ (a ∨ b) = aa ∨ (a ∧ b) = a
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Poset - Lattice Relationship
If P is a poset, then < P, inf , sup > is a lattice.
a ∧ b = inf (a, b)a ∨ b = sup(a, b)
Every lattice is a poset where these definitions hold.
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Poset - Lattice Relationship
If P is a poset, then < P, inf , sup > is a lattice.
a ∧ b = inf (a, b)a ∨ b = sup(a, b)
Every lattice is a poset where these definitions hold.
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Duality
Definition
dual:∧ instead of ∨, ∨ instead of ∧
Theorem (Duality Theorem)
Every theorem has a dual theorem in lattices.
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Duality
Definition
dual:∧ instead of ∨, ∨ instead of ∧
Theorem (Duality Theorem)
Every theorem has a dual theorem in lattices.
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Lattice Theorems
Theorem
a ∧ a = a
Proof.
a ∧ a = a ∧ (a ∨ (a ∧ b))
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Lattice Theorems
Theorem
a ∧ a = a
Proof.
a ∧ a = a ∧ (a ∨ (a ∧ b))
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Lattice Theorems
Theorem
a � b ⇔ a ∧ b = a ⇔ a ∨ b = b
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Lattice Examples
Example
< P{a, b, c},∩,∪ >
⊆ relation
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Bounded Lattice
Definition
lower bound of lattice L: 0∀x ∈ L 0 � x
Definition
upper bound of lattice L: I∀x ∈ L x � I
Theorem
Every finite lattice is bounded.
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Bounded Lattice
Definition
lower bound of lattice L: 0∀x ∈ L 0 � x
Definition
upper bound of lattice L: I∀x ∈ L x � I
Theorem
Every finite lattice is bounded.
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Bounded Lattice
Definition
lower bound of lattice L: 0∀x ∈ L 0 � x
Definition
upper bound of lattice L: I∀x ∈ L x � I
Theorem
Every finite lattice is bounded.
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Distributive Lattice
distributive lattice:
∀a, b, c ∈ L a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)∀a, b, c ∈ L a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ c = c
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ c = c
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ c = c
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ c = c
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ c = c
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ c = c
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ c = c
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ I = I
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ I = I
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ I = I
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ I = I
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ I = I
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ I = I
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Counterexamples
Example
a ∨ (b ∧ c) = a ∨ 0 = a(a ∨ b) ∧ (a ∨ c) = I ∧ I = I
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Distributive Lattice
Theorem
A lattice is nondistributive if and only if it has a sublatticeisomorphic to any of these two structures.
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Join Irreducible
Definition
join irreducible element:a = x ∨ y ⇒ a = x ∨ a = y
atom: a join irreducible elementwhich immediately succeeds the minimum
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Join Irreducible
Definition
join irreducible element:a = x ∨ y ⇒ a = x ∨ a = y
atom: a join irreducible elementwhich immediately succeeds the minimum
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Join Irreducible Example
Example (divisibility relation)
prime numbers and 1 are join irreducible
1 is the minimum, the prime numbers are the atoms
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Join Irreducible Example
Example (divisibility relation)
prime numbers and 1 are join irreducible
1 is the minimum, the prime numbers are the atoms
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Join Irreducible
Theorem
Every element in a lattice can be writtenas the join of join irreducible elements.
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Complement
Definition
a and x are complements:a ∧ x = 0 and a ∨ x = I
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Complemented Lattice
Theorem
In a bounded, distributive latticethe complement is unique, if it exists.
Proof.
a ∧ x = 0, a ∨ x = I , a ∧ y = 0, a ∨ y = I
x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y)
= x ∨ y = y ∨ x = I ∧ (y ∨ x)
= (y ∨ a) ∧ (y ∨ x) = y ∨ (a ∧ x) = y ∨ 0 = y
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Topics
1 Algebraic StructuresIntroductionGroupsRings
2 LatticesPartially Ordered SetsLatticesBoolean Algebra
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Boolean Algebra
Definition
Boolean algebra:< B,+, ·, x , 1, 0 >
a + b = b + a a · b = b · a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)a + 0 = a a · 1 = aa + a = 1 a · a = 0
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Boolean Algebra
Definition
Boolean algebra:< B,+, ·, x , 1, 0 >
a + b = b + a a · b = b · a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)a + 0 = a a · 1 = aa + a = 1 a · a = 0
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Boolean Algebra
Definition
Boolean algebra:< B,+, ·, x , 1, 0 >
a + b = b + a a · b = b · a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)a + 0 = a a · 1 = aa + a = 1 a · a = 0
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Boolean Algebra
Definition
Boolean algebra:< B,+, ·, x , 1, 0 >
a + b = b + a a · b = b · a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)a + 0 = a a · 1 = aa + a = 1 a · a = 0
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Boolean Algebra
Definition
Boolean algebra:< B,+, ·, x , 1, 0 >
a + b = b + a a · b = b · a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)a + 0 = a a · 1 = aa + a = 1 a · a = 0
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Boolean Algebra - Lattice Relationship
Definition
A Boolean algebra is a finite, distributive, complemented lattice.
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Duality
Definition
dual:+ instead of ·, · instead of +0 instead of 1, 1 instead of 0
Example
(1 + a) · (b + 0) = bdual of the theorem:(0 · a) + (b · 1) = b
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Duality
Definition
dual:+ instead of ·, · instead of +0 instead of 1, 1 instead of 0
Example
(1 + a) · (b + 0) = bdual of the theorem:(0 · a) + (b · 1) = b
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Boolean Algebra Examples
Example
B = {0, 1},+ = ∨, · = ∧
Example
B = { factors of 70 }, + = lcm, · = gcd
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Boolean Algebra Examples
Example
B = {0, 1},+ = ∨, · = ∧
Example
B = { factors of 70 }, + = lcm, · = gcd
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Boolean Algebra Theorems
a + a = a a · a = aa + 1 = 1 a · 0 = 0a + (a · b) = a a · (a + b) = a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)
a = aa + b = a · b a · b = a + b
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Boolean Algebra Theorems
a + a = a a · a = aa + 1 = 1 a · 0 = 0a + (a · b) = a a · (a + b) = a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)
a = aa + b = a · b a · b = a + b
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Boolean Algebra Theorems
a + a = a a · a = aa + 1 = 1 a · 0 = 0a + (a · b) = a a · (a + b) = a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)
a = aa + b = a · b a · b = a + b
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Boolean Algebra Theorems
a + a = a a · a = aa + 1 = 1 a · 0 = 0a + (a · b) = a a · (a + b) = a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)
a = aa + b = a · b a · b = a + b
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Boolean Algebra Theorems
a + a = a a · a = aa + 1 = 1 a · 0 = 0a + (a · b) = a a · (a + b) = a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)
a = aa + b = a · b a · b = a + b
![Page 168: Discrete Mathematics - Algebraic Structures](https://reader036.vdocument.in/reader036/viewer/2022081720/5497dd94ac79590e2e8b54dc/html5/thumbnails/168.jpg)
Boolean Algebra Theorems
a + a = a a · a = aa + 1 = 1 a · 0 = 0a + (a · b) = a a · (a + b) = a(a + b) + c = a + (b + c) (a · b) · c = a · (b · c)
a = aa + b = a · b a · b = a + b
![Page 169: Discrete Mathematics - Algebraic Structures](https://reader036.vdocument.in/reader036/viewer/2022081720/5497dd94ac79590e2e8b54dc/html5/thumbnails/169.jpg)
References
Required Reading: Grimaldi
Chapter 7: Relations: The Second Time Around
7.3. Partial Orders: Hasse Diagrams
Chapter 15: Boolean Algebra and Switching Functions
15.4. The Structure of a Boolean Algebra