tm matematika x ipa 8
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LOGIC (MATHEMATICAL LOGIC)
Mathematical logic (also symbolic logic, formal logic, or, less frequently, modern logic) is a subfield
ofmathematics with close connections to the foundations of mathematics, theoretical computer
science and philosophical logic. The field includes both the mathematical study of logic and the
applications of formal logic to other areas of mathematics. The unifying themes in mathematical logicinclude the study of the expressive power offormal systems and the deductive power of
formal proofsystems.
In the mathematical logic, there are two sentences i.e..
1. Close sentence (statement)2. Open Sentence
Explanation
Statement (close statement) is a sentence that only has true value or false value, but not all atonce true or false.
Open sentence is a sentence that has not yet determined whether the value is only true or isonly false.
Interrogative and imperative are not included neither statement nor open sentence
Adriand Nata Kusumah
http://en.wikipedia.org/wiki/Modern_logichttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Foundations_of_mathematicshttp://en.wikipedia.org/wiki/Theoretical_computer_sciencehttp://en.wikipedia.org/wiki/Theoretical_computer_sciencehttp://en.wikipedia.org/wiki/Philosophical_logichttp://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Formal_systemhttp://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Formal_systemhttp://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Philosophical_logichttp://en.wikipedia.org/wiki/Theoretical_computer_sciencehttp://en.wikipedia.org/wiki/Theoretical_computer_sciencehttp://en.wikipedia.org/wiki/Foundations_of_mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Modern_logic -
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Ex : State whether the following sentences and statements, open sentences, nor neither
one.
If they are statements, determine the truth value.
1. 23 < 32 (statement, true)2. x> 4 (open sentence)3. One week consists of (comprises) seven days. (statement, true)4. x 6 = x + 65. x2 9 = 06. x2+ 9 = 0, x R7. How tall is she? (neither one)8. Welcome (neither one)9. Go out (neither one)
Exponent Inequalities
For a > 1
ax> a
ythen x > y
ax< a
ythen x < y
For o < a < 1
ax> a
ythen x < y
ax< ay then x > y
Agastya Prabhaswara Putra
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Negation (Logic Denial)
Negation of Statement notated with
Table of Negation Truth Value
Ex. Given that is
Determine the negation TV of negation !
Sol. , so
Dania Rahmah Aisyah
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Compound statement
1. From two statements of P and Q can be formed a compound statement in the form p or q which is
called disjunction and notated with p v q.
Table of Disjunction TV
p q p v q
T T T
T F T
F T T
F F F
Dzikry Lazuardi Z. S.
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Compound statement
2 ) From two statements of P and Q can be form a compound statement in the form p and q which is
called conjunction and notated with p q .
Table of Conjunction TV
P Q P QT
T
F
F
T
F
T
F
T
F
F
F
Ex : Det the TV of
Tan 60 > sin 0 and cos 45 = 1
T F = F
Exercises
1 . Determine the component and the TV of statements below !
a) Elbow-angled triangle ABC, but both legs are not the same.
b) Two is a prime number or even number.
c) After graduating from school I would take a course or work.
d) Each prime number is divisible by 1 and itself.
e) Someone who is 17 years old and already married are required to have ID cards.
2 . Find the value of x so the sentence - x = 0 and 89 < 0 become
a ) Conjucntion with false value.
b) Conjunction with true value.
Faisal Rahman Yulistian
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Compound statement
3. From two statements of p and q can be formed a compound statement in the form If p then q
which is called implication and notated with pq.
Table of implication TV:
p q p q
T T T
T F F
F T T
F F T
Ex . Det the TV of
If tan 30 = 1/33 , then cos 30 =1/2
If T then F = F
Farrah Fauziyyah K.
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Compound statement
4. From Two statements of p and q can be formed a compound statement in the form p if only if q
which called Bi-implicationand notated with p q.
Truth Value of Bi-implication.
p q p q
T T T
T F F
F T F
F F T
Conclusion:
- In Bi-implication, we will have the true value if both of the inputs are same.- In Bi-implication, we will have the false value if the inputs are different.
Ihsan Hafiyyan
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Det the TV of the following CS!
1. (p ~q) ~p
Sol.
p q ~p ~q (p ~q) (p ~q) ~qT T F F F T
T F F T T F
F T T F F T
F F T T F T
So, the TV of (p ~q) ~p is TFTT
2. (~p q) ~q
Sol.
p q ~p ~q (~p q) (~p q) ~qT T F F F T
T F F T F T
F T T F T F
F F T T F T
So, the TV of (~p q) ~q is TTFT
3. (~p V ~q)(qp)
Sol.
p q ~p ~q (~p V ~q) (qp) (~p V ~q)(qp)T T F F F T F
T F F T T T T
F T T F T F F
F F T T T T T
So, the TV of(~p V ~q)(qp) is FTFT
Indira Anindyajati Prasetyo
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Negation of Compound Statement
1.~(p v q) = ~p ^ ~q
2. ~(p^q) = ~p v ~q
Sol.
1.
p q ~p ~q p v q ~(p v q) ~p ^ ~q
T T F F T ~(T) = F F
T F F T T ~(T) = F F
F T T F T ~(T) = F F
F F T T F ~(F) = T T
So, the TV of ~p ^ ~q is FFFT, able for ~(p v q)
2.
p q ~p ~q p ^ q ~(p ^ q) ~p v ~q
T T F F T ~(T) = F F
T F F T F ~(F) = T T
F T T F F ~(F) = T T
F F T T F ~(F) = T T
So, the TV of ~p ^ ~q is FFFT, able for ~(p v q)
Irfandi Makmur Putra
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Negation of Compound Statement
3. ~(p q) = p ~q
4. ~(pq) = (p~q) v (~pq)
Proof
p q ~p ~q pq ~(pq) p ~q
T T F F T F F
T F F T F T T
F T T F T F F
F F T T T F F
So, the TV of ~p ^ ~q is FFFT, able for ~(p v q)
Proof
p q ~p ~q p~q ~pq pq ~(pq) (p~q) v (~pq)
T T F F F F T F F
T F F T T F F T T
F T T F F T F T T
F F T T F F T F F
So, the TV of ~p ^ ~q is FFFT, able for ~(p v q)
Lathifah Nurrahmah
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Equivalent of compound statement(~p q)
1. Pq = (pq) (q p)= (~p v q) (~q v p)
2. p q = ~p v q3. ~p q = p v q
Proof number 1
~(Pq) = ( p ~q) v ( ~pq )
~[ ~(Pq) ] =~[( p ~q) v ( ~p q ) ]
Pq = ~(p ~q) ~(q ~p)
= (~p v q) (~q v p)
Proof number 2
~(p q) = p ~q
~ [~(p q)] =~[ p ~q ]
P q = ~p v q
What is the meaning of the following slogan?
Smoke or healthy
Sol. Supposing ~p = smoking and q = healthy
~p v q
Is same as
pq = Doesnt smoking then healthy
Another example :
p q ~p ~q (~p v q) (~q v p) (p q) (q p)
T T F F T T T T
T F F T F T F T
F T T F T F T F
F F T T T T T T
(pq) (q
p)(~p v q) (~q v p) Pq
T T T
F F F
F F F
T T T
1. 2 x 2 = 4 if only if 4 : 2 = 2 the value is trueT T = T
2. 2 x 4 = 8 if only if 8 : 4 = 0 the value is falseT F = F
Translated from :
http://www.matematikamenyenangkan.com/logika-matematika
Mahdiar Naufal
http://www.matematikamenyenangkan.com/logika-matematika/http://www.matematikamenyenangkan.com/logika-matematika/ -
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Implication and Contraposition, Converse and Inverse
A. Implication and ContrapositionProve
p q ~p ~q p q ~q ~p
T T F F T T
T F F T F F
F T T F T T
F F T T T T
B. Converse and InverseProve
p q ~p ~q q p ~p ~q
T T F F T T
T F F T F F
F T T F T T
F F T T T T
M. Ilyas Arradya
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Determine converse, inverse, and contraposition! (State the truth value)
1. If tan 30 =
then cos 30
.
Converse: If cos 30
, then tan 30 =
(T F = F)
Inverse : Iftan 30
, then cos 30=
(T
F = F)Contraposition: If cos 30=
, then tan 30
(F T = T)2. If sin 0 < cos 0 then cosec 30 = 2
Converse: If cosec 30 = 2, then sin 0 < cos 0 (T T = T)
Inverse: If sin 0 > cos 0, then cosec 30 2 (F F = T)
Contraposition: If cosec 30 2, then sin 0 > cos 0 (F F = T)
3. If Persib doesnt win then bobotoh are sadConverse: If bobotoh are sad ,then Persib doesnt win (F F = T)
Inverse: If Persib win, then bobotoh are happy (T T = T)
Contraposition: If bobotoh are happy, then Persib win (T T = T)
M. Imam Nasrullah
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Tautology, Contradiction, and Contingency
Exp.
Tautology is a compound statement which the value is always true (all T)
Contradiction is a compound statement which the value is always false (all F)
Contingency is a compound statement which the value is not always true or not always false.
Contingency is not Contradiction nor Tautology
P ~P P ~P P ~P P ~P P ~P
T F T F F F
T F T F F F
F T T F T FF T T F T F
BO the TV of (P V ~P ) is always true ( all true ), then (P V ~P ) is Tautology
BO the TV of (P ~P ) is always false ( all false ), then (P ~P ) is Contradiction
BO the TV of (P ~P ) is not always true or not always false, then (P~P ) is Contingency
BO the TV of (P ~P ) is always false ( all false ), then (P~P ) is Contradiction
M. Nur Fathurrahman
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State/ add the follc.s. are tautology, contra diction, or contingency
1. (p ^ q) ->q = Tautology2. p -> (p v q) = Tautology3. (p^q) -> (p v q) = Tautology
4. (p -> q) ^(p ^~q) = Contradiction
Sol.
1. Table of TV from (p ^ q) -> q
p qp^ q (p ^ q)-> q
T T T T
T F F T
F T F T
F F F T
For (p ^ q) -> q Because of (p ^ q) -> q is always true (all true), then
(p ^ q) -> q is Tautology
T
T
T
T
M. Raditya Dwiprasta
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2| For p (p V q)
p q p V q p (p V q)
T T T T
T F T T
F T T TF F F T
Because of TV of p ( p V q) is always true, then p (p V q) is tautology
3| For (p q) (p V q)
p q p q p V q (p q) (p V q)
T T T T T
T F F T T
F T F T T
F F F F T
Because of the TV of (p q) (p V q) is always true, then (p q) (p V q) is tautology
4| For (pq) (p~q)
p q ~q pq p ~q (pq) (p ~q)
T T F T F F
T F T F T FF T F T F F
F F T T F F
Because of the TV of (pq) (p ~q) is always false, (pq) (p ~q) is contradiction
By: M. Umar Fathurrohman
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Infrence ( drawing conclution)
Ponens modus
Premise I : p q
Premise II: p
Conclution: q
Ex: if Im diligent then Im clever
Im diligent
Conclution Im clever
This argument is valid (prove)!
If it is stated in implication form [(p q) ^p] q and its the tautology
Nabila Putri Fauzia
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Proof
p q p q (p q) ^p [(p q) ^p q
T T T T T
T F F F T
F T T F TF F T F T
BO the TV of [(p q) ^p] q is always true ( all T ) or its tautology, then the implication is valid.
Nadia Gitta Paramita
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Infrence ( drawing conclution)
Tollens Modus
Premise I : pq
Premise II : ~q
Conclusion : ~p
Example
If Im diligent then Im clever
Im stupid
Conclusion Im lazy
This argument is valid (prove)!
Nadira Nurul F.
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Infrence ( drawing conclution)
If it is stated in the implication , - , Tollens Modus as said valid if the is of, - is ..
Proof
, -
F F T T T F T
F T T F F F T
T F F T T F T
T T F F T T T
BO the TV of , - is always True (all T) and the implication statement of, - is valid.
Naufal Purnama Hadi
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Infrence ( drawing conclution)
Syllogism
Premise I : pq
Premise II : qr
Conclusion : pr
Ex. PI : If Im diligent then Im clever
PII : If Im clever then Im successful
Conclusion: If Im diligent then Im successful
Even number :Bil. Genap
Odd number :bil. Ganjil
Integer: bil. Bulat: {..,-1,0,1,..}
Natural Number: {1,2,3,}
Whole Number: {0,1,2,}
Real Number: {..1,..,-,..0,..,
,..1,..}
Prime Number: {2,3,5,..}
Complex:
Putri Egayulia N.
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This argument is valid (Prove! )
If its stated in the implication form is [(pq)(qr)(pr)]
Syllogism is said valid if the is of *(pq)(qr)(pr)] is tautology
Proof
p q r pq qr pr [(pq)(qr) [(pq)(qr)(pr)]
T T T T T T T T
T T F T F F F T
T F T F T T F T
T F F F T F F T
F T T T T T T T
F T F T F T F T
F F T T T T T T
F F F T T T T T
Because Of the TV of [(pq)(qr)(pr)+ is always true (all T) or its tautology, that the implication
statement of [(pq)(qr)(pr)] is valid !
Rr. Audria Pramesti W.
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Quantor / Quantifier statements
There are two i.e.
1. Special Quantifier (Existentian Quantor): it is symbolized with (read : there is/ are), (read : so that).Notation : x S P(x)
There is x an element of S, so that P(x) is valid (holds)
Ex : Det. The TV of the foll sq
1. x R x 5 = 8Sol. (there is x an element of real number, so that x + 5 = 8 is valid)
BO there is x an element of real, that is 3, so that if x is changed with 3, then the
statement above becomes 3 + 5 = 8 is true.
So, the tv of the sq of x R x 5 = 8 is true.
Rasya Salma Irawan
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Quantor / Quantifier statements
2. X R x2 >0
Sol. ( There is x an element of real number, so that x2> 0 is not valid
Bo there is x an element of real, that is , so that if as with 3, then the statement above becomes 32> 0
as true. So the TV of the Sq of x x2> 0 as true
3. x R x2< 0
Sol BO there is not x value an element of real which causes x2< 0
So, x R x2< 0 as false
2 General Quantifier ( universal Quantor)
It is symbolized with (read:for each/for all)
Notation : x S p(x)
(for each/for all x an element of s, so that p(x) is valid
Ex: Det the TV of the fall qq
1. x I 2x = 1
Reza Fasya
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Solution
Because of there is not x value an element an integer which causes 2x=1 is true,
So, x I 2x = 1 is false
2. x R x2=0
Solution
Because of there is value x of an element real that is zero, so that if x changed with 0, then the
statement above becomes 02=0 is false
So, x R x2=0 is false
3. x R x2+9=0
Solution
There is not x value an element of real causes x2+9=0 is true
So, the TV of x R x2+9=0 is false
Negation of Quantifier Statement
1. ~[ x S P(x)] is x S ~P(x)2. ~[ x S P(x)] is x S ~P(x)
Ex/Exercise
Determine the negation and the TV of the following q.s
1. x R x>2x2. x R x=03. x x2-404. x W x-2
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Sol.
1. x R x > 2xBO there is x value of an element of real, that is -1, so that if x changed with -1, then the
statement above becomes -1>-2 is true.
So, x R x > 2x is true.~[ x R x > 2x+ is x R x 2x
BO there is x value of an element of real, that is -1, so that if x changed with -1,then the
statement above becomes -1-2 is false.
So, x R x 2x is false.
2. x R = 0BO there is x value of an element of real, that is 0, so that if x changed with 0, then the
statement above becomes = 0 is true.So, x R = 0 is true.~[ x R = 0] is x R 0BO there is x value of an element of real, that is 0, so that if x changed with 0, then the
statement above becomes 0 is false.So, x R 0 is false.
3. x x24 0BO there is x value of an element of complex, that is , so that if x changed with , thenthe statement above becomes ()24 0 is true.So, x x24 0 is true.
~[ x x24 0+ is x x2 4 = 0BO there is x value of an element of complex, that is , so that if x changed with , thenthe statement above becomes ()2 4 = 0 is false.So, x x2 4 = 0 is false.
4. x W x 2 < -3 = false~[x W x 2 < -3] = x W x2 -3 = true
Shazkia Aulia S. D.
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Exponential Inequalities
1.
2.
3. ()
* +
Sri Utami Ayuningrum
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Logarithms Inequalities
Logarithms Function
Logarithms function is inverse of exponential function.
Proof F.E: Logarithm function y=f(x)= logax is inverse of exponent function y=f(x)= ax
y = ax log y= log axlog y = x log a
x = log y / log a
x = logay
= loga y
= loga x
Syifaulqulub A. Nurfahdani
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Graph : 2. Y = f(x) = 2xand Y = f(x) = log x
1. Y = f(x) = log x and Y =f(x) =
For y =logx
If x1 > x2 thenlogx1
logx2
If x > y thenalogx
alog y then x < y
Det.2log(x
2 2x ) > 3
2log(x
2 2x) >
2log8
x2 2x> 8
x2 2x 8 > 0
Zp : x2 2x 8 = 0
(x 4)(x + 2) = 0 + + + - - - + + +
x = 4, x = -2 -2 4
x > 4
Condition :
x2 2x > 0
Zp : x (x 2) = 0 + + + - - - + + +
x = 0, x = 2 0 2 4
+ + + - - - + + +
-2 0 2 4
x < -2 V x > -4
Thalia Nurul H.
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Note: y = logax a > 0 , a 1 , x > 0
x ... -3 -2 -1 0 1 2 3 ... 1 2 4 8
Y = f(x) = 2x
1 2 4 8
Y = f(x) = log x -3 -2 -1 0 1 2 3
Y =f(x) =
0 -1 3 2 1 2 3
X = 2 y = log x = 1
X = 4 y = log x = 2
X = 8 y = log x = 3
If x > x then loga x > logax
x < x then loga x < logax
If x > y , then loga x >loga y
If loga x > loga y , then x > y , a > 1
Veby Virgiana
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Keterangan
Adriand Nata Kusumah : Meaning of Statement and Open Sentence
Agastya Prabhaswara Putra : Exercises 1 + Exponent Inequalities
Dania Rahmah Aisyah : Negations
Dzikry Lazuardi Zammuruddan Soeharno : Disjunction
Faisal Rahman Yulistian : Conjunction
Farrah Fauziyyah Kurniawaty : Implication
Ihsan Hafiyyan : Bi-Implication
Indira Anindyajati Prasetyo : Exercises 2
Irfandi Makmur Putra : Negation of Compound Statement
Lathifah Nurrahmah : Negation of Compound Statement
Mahdiar Naufal : Equivalent of compound statement
Muhammad Ilyas Arradya : Converse, Inverse, and Contradiction + Editor
Muhammad Imam Nasrullah : Exercises 3
Muhammad Nur Fathurrahman : Tautology, Contradiction, and ContingencyMuhammad Raditya Dwiprasta : Exercises 4
Muhammad Umar Fathurrohman : Exercises 4
Nabila Putri Fauzia : Ponens Modus + Editor
Nadia Gitta Paramita : Ponens Modus
Nadira Nurul Fadhilah : Tollens Modus
Naufal Purnama Hadi : Tollens Modus
Putri Egayulia N. : Syllogism
Raden Roro Audria Pramesti Wulandari : Syllogism
Rasya Salma Irawan : Special Quantifer
Reza Fasya : General Quantifer
Rissa Zharfany E. : General Quantifer
Shazkia Aulia Shafira Dewi : Exercises 5
Sri Utami Ayuningrum : Exponential Inequalities
Syifaulqulub Adina Nurfahdani : Logarithms Inequalities
Thalia Nurul Heraswati : Exercises 6
Veby Virgiana : Exercises
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