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    1

    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