Logic Math .………………………………………………………………….. Hal : 1

CHAPTER VLOGIC MATH

A. GOALS FOR UNIT V1. I will know the definition of sentences of proposition, open sentence and

sentences of non-proposition.2. I will know the symbol of propositional logic3. I will be able to determine the negation of proposition4. I will be able to determine the value and the truth table of conjunction5. I will be able to determine the value and the truth table of disjunction6. I will be able to apllicate conjunction and disjunction in circuit diagram7. I will be able to determine the value and the truth table of implication8. I will be able to determine the value and the truth table of bi implication9. I will be to use De Morgan’s Duality Law10. I will be able to determine the negation of implication11. I will be able to determine the negation of bi implication12. I will know the definition of tautology, contradiction and contingency13. I will know the definition of Quanterized Proposition14. I will be able to write the symbol of Universal Quainter15. I will be able to determine the negation of Universal Quainter16. I will be able to write the symbol of Existential Quainter17. I will be able to determine the negation of Existential Quainter18. I will be able to determine Converse, Inverse and Contraposition from Implies

proposition19. I will be able to draw conclution using direct proofs20. I will know the step to proof math using indirect proofs21. I will know the step to proof math using the Principal of Mathematical Induction22. I will able to proof math using mathematical Induction

B. WORDS TO REMEMBERPropositionOpen sentencesSet solutionCircuit diagramNegation

Switching networkSerial circuitParallel circuitAntecedent Consequence

ConclutionPremiseQuainterArgumentValid

C. CONCEPT

In this unit, we study meaningful sentences, they are :1. Proposition

is a kind of a sentence which has a true OR false value, but not a true AND false one.The truth value depends on the truth or falsity of stated reality. Where as the truth or the falsity of a proposition is called the truth value of the proposition

2. Open Sentencesare sentences which contain variables and will be propositions if the variables are changed into constants in their universal setsExample 1 : x – 5 23 x2 – 2x + 1 = 0 4x + 5 = 40

3. Non PropositionA kind of a sentence which haven’t a true or false value

Example 2 :

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a. 2 is a prime number ( Proposition )b. Please, have a sit ! ( Not Proposition )c. Mount Jayawijaya is in Aceh ( Proposition )d. 3x2 + 8x + 5 < 0 ( Open sentence )

THE SYMBOL OF PROPOSITION LOGIC The symbols of certain propositions usually use “proposition variables” , like p, q or r

a. Single Propositionp : I live in 15 Blimbing Streetq : I study in State Junior High School

b. Compound PropositionI study in State Junior High School AND live in 15 Blimbing StreetMy mother buys fruits OR vegetables

There are 5 proposition operators and it can observe the table below

NO OPERATORS DAILY LANGUAGENAME SYMBOL1. Negation ~ No, not2. Conjunction And, but, although, nevertheless3. Disjunction Or 4. Implication If ……. so …….5. Bi implication If and only if …… so ….. if

NEGATION OF A PROPOSITIONIf p is a proposition, then the negation of it, is put in notation as ~p or . Hence, if the proposition of p is true, then the proposition of ~p is false, and vice versa.Example 3 :a. p : Lima is the capital city of Singapore

~p : Lima is not the capital city of Singapore~p : It’s not true that Lima is the capital city of Singapore

b. q : 5x – 2 = 3~q : 5x – 2 3

c. r : 2x – 5 13~r : 2x – 5 13

The table of negation truth valuesp ~p T FF T

Note : T is TrueF is False

D. PROBLEM SOLVING

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1. State the following sentences proposition, not proposition or open sentences !a. 23 is a rational numberb. 6x – 3 = 9c. 63 is divided by 8d. 39 is an even number

e. what is your father ?f. Let’s go !g. 3x – 9 > 18

Solution :a. …………………..b. …………………..c. …………………..d. …………………..

e. …………………..f. …………………..g. …………………..

2. Determine the set solution from the problem below !a. 2x – 1 < 5 , x N

Solution :x = 1 2.1 – 1 < 5 1 < 5x = 2 ………… ………….x = 3 ………… ………….x = 4 ………… ………….

SS = { ………………… }

b. x2 – 6x + 5 = 0, x NSolution :………………………………………………………………………………………………………………………………………………………………

3. Determine the negation of the following propositions and determine the truth values !a. Palembang is the capital city of South Sulawesib. sin 210 has positive valuec. There are 12 month in a yeard. Six is not a count numbere. x2 – 6 9Solution :a. ………………………b. ………………………c. ………………………

d. ………………………e. ………………………

E. EXERCISE 1

I. Complete the following blanks !1. The truth value of “2n always an even number” is ……………………………….2. The truth value of “cos 75 complement with sin 15” is ………………………...3. The solution set of 2x2 – x – 3 0 are

…………………………………………….4. Negation of “ is not a root form” is…………………………………………5. Negation of “ 4a + 62 = 0 is a quadratic” is ………………………………………

II. Choose the right answer of the following questions !1. From proposition below that have truth value is ………..

a. This pizza is deliciousb. 11 + 7 20c. Taj Mahal is in Agra city

d. 5 + 2 0e. Borobudur temple is in Solo

2. From sentences below that an open sentences is ………..a. 2 + 5 = 8 b. 13 + 3 13

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c. The rose is bloomingd. 2x – 2 10

e. Michael Jackson is a farmer

3. From proposition below that have false value is …………a. 2 is a prime numberb. 3 + 7 9c. 2n is an even number

d. The Sun rises in Easte. Zambia is in Africa

4. The negation of “All farmers are men” is …………..a. All farmers are not womenb. There is a woman farmerc. Nor all farmers are women

d. All farmers are womene. There is a man farmer

5. The set solution from x2 – 6x + 5 0 is ………….a. { x x 5 or x 6 }b. { x x 1 or x 5 }c. { x x -5 or x 1 }

d. { x x -5 or x 6 }e. { x -5 x 1 }

6. Negation of “3 is a count number” is ……………a. It’s not true that 3 is a count numberb. 3 less than 5c. 2 is not a count numberd. 3 is not a prime numbere. 3 is not a rational number

7. Negation of “There is x N so that x 7” is …………..a. For all x N, x 7b. For all x N, x 7c. For all x N, x 7

d. For all x N, x 7e. There is x N, x 7

8. Negation of “ 5x2 + 8x + 3 0 is …………….a. { x x 1 or x }

b. { x x -1 or x }

c. { x x - or x -1 }

d. { x - x -1 }

e. { x -1 x }

III.Solve the problem below !1. Determine the solution set of “ -3x2 + 3x + 6 0” !

Solution :……………………………………………………………………………………..……………………………………………………………………………………..

2. Determine the truth value of “All x2 = c, c R, have a real root” !Solution :…………………………………………………………………………………………………………………………………………………………………………

3. Determine negation of “Mongolia is in Asia” !Solution :……………………………………………………………………………………...…………………………………………………………………………………….

C. CONCEPT

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CONJUNCTIONSare compound propositions which have and as the operator. Two propositions of p and q which are stated in the form of “p q” are called conjunctions and are read p and q. The conjunctions and often means “then, next”. Conjunctions of two propositions of p and q will be true if only the two components are true.

The table of conjunction truth values :

P q p qT T TT F FF T FF F F

The word “and” the same meaning with but, although, whereas, while, that, also, etc. Symbol of “” use to define the intersection of two sets.A B = { x x A and x B }

Example 4 :Determine the truth value for each propositions below !a. Jakarta is the capital city of Indonesia and 2 + 3 = 5b. 53 is prime number and all prime numbers are odd numbersSolution :a. Jakarta is the capital city of Indonesia ( T )

2 + 3 = 5 ( T )Hence, Jakarta is the capital city of Indonesia and 2 + 3 = 5 have true value

b. 53 is prime number ( T )All prime numbers are odd numbers ( F )Hence, 53 is prime number and all prime numbers are odd numbers have false value

Example 5 :Determine the value of x to make the following propositions have TRUE value !a. 6 is even number and 3x – 5 2x + 4b. 51 is divided by 3 and 3log x = -4

Solution :a. Let p : 6 is even number

q : 3x – 5 2x + 4“p q” become true if both of them are truep : 6 is even number ( T )q : 3x – 5 2x + 4 ( it must be true )So, 3x – 5 2x + 4 3x – 2x 5 + 4 x 9 6 is an even number and 3x – 5 2x + 4 will be true if the value of x 9

b. Let p : 51 is divided by 3q : 3log x = -4“pq” become true if both of them are truep : 51 is divided by 3 ( T )

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q : 3log x = -4 ( it must be true )So, 3log x = -4

3-4 = x x =

51 is divided by 3 and 3log x = -4 will be true if the value of x =

DISJUNCTIONSare compound propositions which have or as the operator. Two propositions of p or q which are stated in the form of “p q” are called disjunctions and are read p or q. Disjunctions of two propositions of p or q will be true if one or both of propositions p and q are true.

There are two kinds of disjunction :1. Exclusive Disjunctions

A Disjunctions which states other component can be true can be false. Two propositions can be p only is true, q only is true, but neither or false for both of p and q are true. The notation is “p q” and this exclusivity is often emphasized by using the word “one of”

Example 6 :I was born in Tegal or Padang

This disjunctions means that it just choose one of them, Tegal or Padang, but not both of them. Because, none in the world who was born in 2 places at the same time.

The table of Exclusive Disjunctions truth values :

P q p qT T FT F TF T TF F F

2. Inclusive DisjunctionsA Disjunctions that is p q will be true if one or both of propositions p and q are true.

Example 7 :3 is a natural number or count number

This disjunctions means that it can be use for both of them, natural number or count number; natural number and count number.

The table of Inclusive Disjunctions :

P q p qT T TT F T

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F T TF F F

For the next, it just discuss about Inclusive Disjunctions.

Example 8 :Determine the truth value of each propositions below !a. Bandung is the capital city of West Java or 5 + 3 = 7b. 2 is a prime number or

Solution : a. ……………………………………………………………………………………

……………………………………………………………………………………b. ……………………………………………………………………………………

……………………………………………………………………………………

Example 9 :Determine the value of x to make the following propositions have FALSE value !a. 2x – 5 = 10 or a equilateral triangle have different sides

b. 8 = or x2 – 5x – 6 = 0

Solution :a. ……………………………………………………………………………………

……………………………………………………………………………………b. ……………………………………………………………………………………

……………………………………………………………………………………

APLICATION CONJUNCTION AND DISJUNCTION IN CIRCUIT DIAGRAM

One of logic system application in daily life is switching network.

Switch is open

It means that there’s no electricity current ( off )

Switch is closed

It means that there is electricity current ( on )

A. Serial Circuit

A p q B

Figure 1.1

Figure 1.1 above shows a serial circuit. Electricity current from A to B will flow if only switch p and switch q are on together. Symbol of p q is to state that A and B are connected in serial way. So, serial circuit has a conjunction characteristics.

The truth table of Conjunction Circuit

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p q p q1 1 11 0 00 1 00 0 0

B. Parallel Circuit p

A B

q

Figure 1.2

Figure 1.2 above shows a parallel circuit. Electricity current from A to B will flow if only on or both switches p and q are ON. Symbol of p q is to state that A and B are connected in a parallel way. So, parallel circuit has a disjunction characteristics.

p q p q1 1 11 0 10 1 10 0 0

D. PROBLEM SOLVING

1. Let p : 2 is a prime numberq : 2 is an even number

Write the logic symbol for each propositions below !a. 2 is not a prime number and an even numberb. 2 is a prime number and not an even numberc. 2 is not a prime number and not an even numberd. 2 is an even number or a prime numbere. 2 is not an even number or not a prime numberf. 2 is an even number or not a prime number

Solution :a. ………………………….b. ………………………….c. ………………………….

d. ………………………….e. ………………………….f. ………………………….

2. Let p is a false proposition and q is true proposition.Write the truth value for the following proposition !

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a. ~pb. ~qc. ~p q

d. p ~qe. ~p ~qf. ~p ~q

Solution :a. ………………………….b. ………………………….c. ………………………….

d. ………………………….e. ………………………….f. ………………………….

3. Complete the following tables !

p q r ~p ~q ~r (p q) ( p q) r (~p ~q) ~rT T T …. …. …. …. …. ….T T F …. …. …. …. …. ….T F T …. …. …. …. …. ….T F F …. …. …. …. …. ….F T T …. …. …. …. …. ….F T F …. …. …. …. …. ….F F T …. …. …. …. …. ….F F F …. …. …. …. …. ….

4. Complete the following tables !

p q ~p ~q ~p q p ~q ~p ~qT T …... …... …... …... …...T F …... …... …... …... …...F T …... …... …... …... …...F F …... …... …... …... …...

5. Determine the truth value from the following propositions !a. 11 or 15 is divided by 3b. sin2x + cos2x = 2 or 1 + tan2x = sec2xc. -5 -1 or 8 is an old numberd. 5log 125 = 3 and 33 = 26e. Irrational numbers are surds and f. (53)2 = 56 or 37.38 = 356

Solution :a. …………………………b. …………………………c. …………………………

d. …………………………e. …………………………f. …………………………

E. EXERCISE 2

I. Complete the following blanks !1. The truth value of “Two days after Sunday is Wednesday or 2 + 4 = 10” is ……..2. Let p : Dean studies Physic

q : Dean studies Math r : Dean studies Chemistry

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The symbol from proposition “Dean studies Physic or Math, but not studies Chemistry” is …………

3. 2x2 + 10x + 8 0 and 31 is prime numberThe value of x in order to make the proposition nave TRUE value is ………….

4.

A p q BSymbol from circuit above is ……………...

5. 3x – 1 = 8 or sin 90 = 1The value of x in order to make the proposition have FALSE value is ……….

II. Choose the right answer of the following questions !1. The truth value of p ~(q p) is …………

a. FFFTb. FFTF

c. FFFFd. FTFF

e. FTTF

2. ~(p ~q) have the same value with …………a. ~p qb. ~p ~q

c. p qd. p ~q

e. ~p ~q

3. Negation of “There are people who don’t have sins” is …………..a. Some people have sinsb. All people do not have sinsc. No people have sins

d. All people have sinse. People and sins

4. Negation of “It’s not true that today is not rain” is …………..a. Today is drizzleb. Yesterday was hard rainc. Tomorrow will hard rain

d. Today is raine. Today is not rain yet

5. The following proposition that include of conjunction, except ………….a. 242 = 576 and Indonesia is a Republic Nationb. Budi does not have a car although he is richc. Although Iwan doesn’t study, he can graduates from schoold. Ali is Teacher’s son but he’s veru stupide. g and l are parallel lines or intercept in one point

6. In order to make “2x + 3 5x – 2, x R and 9 is factor of 72” have TRUE value, then x = …………...

a. x

b. x

c. x

d. x

e. x

7. The value of x in order “3log x = -4, x N or 7 5” have TRUE value is ………a. x = 81b. x -81

c. x = d. x e. x 81

8. Let p is a false value, q is a true value and r is true value. The following symbol have a false value, except ………….a. ( q p ) ~rb. ( p q ) rc. ( q p ) ~r

d. ~( p ~q ) re. ( r q ) p

9. The logic symbol of the following circuits is …………..

p q

r a. ( p q ) rb. ( p q ) rc. ( p q ) r

d. ( ~p ~q ) re. ( ~p q ) r

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10. Symbol of the truth table is ………

p q …….T T FT F TF T FF F F

a. ~p qb. ~p q

c. p ~qd. ~p ~q

e. ~p ~q

11. The truth value from the following table are …………

p q r [( p ~q ) ~r]T T T ……T T F ……T F T ……T F F ……F T T ……F T F ……F F T ……F F F ……

a. F T T F T F T Fb. F T T T F T F Tc. T F F F T F T F

d. T F T F T F T Fe. T T T T T T T F

12. The logic symbol of the following circuits is ………...

p q

r

~qa. ( p q ) ( r ~q )b. ( p q ) ( r ~q )c. ( p q ) ( r ~q)

d. ( q r ) ( p ~q )e. ( p ~q ) ( r p )

13. From the following proposition that have false value is ……….a. 1000 is divided by 6 or 12 is factor of 6b. Frog is an amphibi animal and tiger is a wild animalc. Magelang is in Central Java although Madiun is in East Javad. 6 is an odd number and 2 is a prime numbere. 2 x 3 = 7 or Surabaya is in East Java

14. Symbol of the truth table is ………

p q …….

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T T FT F FF T FF F F

a. ~p qb. p q

c. p ~qd. ~p ~q

e. ~p ~q

15. Symbol of the truth table is …………...

p q r ……….T T T FT T F FT F T FT F F FF T T TF T F TF F T FF F F T

a. ( p q ) r b. p ( q r )c. ( ~p q ) ~r

d. ~p ( q ~r )e. ( ~p ~q ) r

III.Solve the problem below !1. Determine the value of x for the following propositions in order to make it has

TRUE value !a. and 2x – 5 7b. QE : x2 + 1 = 0 have equal roots or c. or 8 is a composit numberd. x2 - 4 0 and x2 – 2x – 3 0

Solution :a. ……………………………………….b. ……………………………………….c. ……………………………………….d. ……………………….........................

2. Let p : The Sun rises q : The day is shinnyWrite the sentences for each symbol below !a. p ~qb. ~(p q)

c. ~p ~qd. ~p q

Solution :a. ……………………………………….b. ……………………………………….c. ……………………………………….d. ……………………….........................

3. Construct the truth table for each propositions below !a. ~p qb. ~(p q)c. (p ~q) ~p

d. (~r q ) ~re. ~r ( p ~q)f. ( ~p ~r) q

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Solution :a. ……………………………b. ……………………………c. ……………………………

d. ……………………………e. ……………………………f. ……………………………

4. Determine the truth value for each propositions below !a. 2log x + 2log y = 2log xy, for x 0 and y 0 and a2 – b2 = ( a + b)(a – b)

b. and 1 is a prime factor of 21

c. 61 is divided by 3 or log 40 – log 4 = log 20

Solution :a. ……………………………………….b. ……………………………………….c. ……………………………………….

C. CONCEPT

IMPLICATIONTwo propositions of p and q which are stated in a sentence of “If p, so q” are called implication / conditional propositions and are symbolized as “p q”. The implication have false value if only p has a true and q has a false value.“p q” is read :

If p so q p implication q q is a necessary requirement for p p is an enough requirement for q

Notes : p is called antecendent; hypothesis; causeq is called conclution; consequence; effect

The table of implication truth value :

p q p qT T TT F FF T TF F T

LOGICAL IMPLICATION

If P and Q are the set solutions of open sentences p(x) and q(x) then p(x) q(x) will be true if P Q

Example 10 :a. If x = 3 so x2 – 9 = 0b. If x 2 so x2 4c. If PQRS is rectangle so P = Q = R = S = 90d. If n is an odd number so ( 2n -1 ) is an odd number

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D. PROBLEM SOLVING

1. Let p is a false proposition and q is a true proposition. Determine the truth value for each compound propositions below !a. p ~qb. ~p ~qc. ~ (~p q)

d. ~ (p q)e. p qf. ~ (~p ~q)

Solution : a. ……………………b. ……………………c. ……………………

d. ……………………e. ……………………f. ……………………

2. Let p : Juleha graduated from Junior High Schoolq : Juleha will get a bicycle from her father

Write the sentences from the following symbol !a. ~q pb. ~p ~qc. ~ (~p q)

d. ~ (p q)e. p qf. ~ (p ~q)

Solution : a. …………………………b. …………………………c. …………………………d. …………………………e. …………………………f. …………………………

3. Complete the tables below !

p q p q ~p ~q ~ (p q) p ~qT T …… …… …… ……T F …… …… …… ……F T …… …… …… ……F F …… …… …… ……

4. Complete the following tables !

p q r ( p q) r ( p q) r ~ ( p q) ~rT T T …… …… ……T T F …… …… ……T F T …… …… ……T F F …… …… ……F T T …… …… ……F T F …… …… ……F F T …… …… ……F F F …… …… ……

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E. EXERCISE 3

I. Complete the following blanks !1. The truth value of “ If log 3 + log 5 = log 8 so 103 + 105 = 108 ” is ……….2. Let p : Kevin passes Math examination

q : Brian gets the prizeThe symbol from proposition “If Kevin doesn’t pass Math examination so Brian doesn’t get the prize” is …………...

3. “If 4x - 5 = 2x -1 so log 5 + log 6 = log 11” The value of x in order to make the proposition have TRUE value is ………….

4. Let p : Grace is hungry q : Grace need foodsSymbol ~ ( p q ) can be written as …………..

5. “If so 1 – 2x = x – 8”The value of x in order to make the proposition have FALSE value is …………..

II. Choose the right answer of the following questions !1. The truth value of ~ (~p q) is …………

a. T T T Tb. T T T F

c. F F F Fd. F T T T

e. T F T F

2. The truth value of ~ (p q) is ………….a. F T F Fb. T F T T

c. F F T Fd. T T F T

e. T T T F

3. ~ (p q) has the same value with …………a. ~p qb. p ~q

c. p ~qd. ~p q

e. q p

4. Let p is a false value and q is a truth value. From the following compound proposition which have a TRUE value is ………..a. p qb. ~p q

c. ~q ~pd. q p

e. ~p ~q

5. Let “If the day is rain so some students will not come late”. The same value with this proposition is …………a. If the day is not rain so same students will not come lateb. If all students are not come late so the day is not rainc. If the day is rain so all students will come lated. If the day is sunny so all students will not come latee. If some students come late so the day is rain

6. Negation of q p is …………a. ~q pb. ~q ~p

c. ~q pd. ~p q

e. p ~q

7. The value of x in order to make the proposition : “If x2 – 3x = 0 so 7 is an even number” has a false value is ………..a. 2 or 3b. 3

c. 1 or 3d. 1

e. 1 or -3

8. Symbol of the truth table is ……………….p q …….T T FT F TF T T

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F F T

a. p ~qb. ~p q

c. p qd. q p

e. ~q ~p

9. Symbol of the truth table is ………………..p q r ………..T T T TT T F FT F T TT F F TF T T TF T F FF F T TF F F F

a. (q r) pb. (p r) p

c. (p q) rd. p (r q)

e. r (q p)

10. Symbol of the truth table is ………………..p q …….T T TT F TF T TF F F

a. (~p q) ~qb. p qc. ~p q

d. ~q ~pe. (~p q) p

III.Solve the problem below !1. Determine the truth value for each proposition below !

a. If x2 is old number so x is a nature number

b. If so

c. If cos 90 = 1 so sin 270 = -1d. If log 25 – log 5 = log 5 so

e. If so 3log

Solution :a. …………………………….b. …………………………….c. …………………………….d. …………………………….e. …………………………….

2. Determine the value of x in order to make each propositions below have a TRUE value !

a. If 3log so x3 – 1 = 0

b. If tan 390 = so

c. If 1 – 3x 4 so 2 is a composit number

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d. If 2x2 – 3x – 5 0 so 72 = 49

Solution :a. …………………………..b. …………………………..c. …………………………..d. …………………………..

3. Let p is a true value, q is a false value and r is a false value. Determine the truth value for each propositions below !a. p ( q r)b. (p q) ~rc. (~r q) ~p

d. (r ~p) qe. (~r ~p) qf. ( q ~r) (p ~q)

Solution :a. …………………b. …………………c. …………………

d. …………………e. …………………f. …………………

4. Construct the truth table for each propositions below !a. q pb. p (r q)c. (p q) (p ~q)d. (p q) (~q ~p)

Solution :a. …………………..b. …………………..

c. ………………….d. ………………….

5. Let x is variable in Real Number. Determine the truth value for each propositions below !a. If x – 3 = 4 so x2 – 5x – 14 = 0b. If x2 – 16 = 0 so x – 4 = 0c. If x2 4 so x 2d. If 2x2 + x – 1 = 0 so 2x – 1 = 0e. If ABC is a isosceles triangle so AB = ACf. If PQRS is a rhombic so PQ = RS

Solution :a. ……………….b. ……………….c. ……………….

d. ………………..e. ……………….f. ……………….

C. CONCEPT

BI IMPLICATIONTwo proposition of p and q which are stated in the symbol of “p q” are called Bi implication / double condition propositions / equivalent.

“p q” is read : p if and only if q p is a necessary and enough requirement for q q is a necessary and enough requirement for p Abbreviation of p q and q p

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Two compound propositions are equivalent if and only if p and q have the same truth value.

The table of Bi implication truth value :

p q p qT T TT F FF T FF F T

LOGICAL BI IMPLICATIONIf p(x) and q(x) are open sentences then p(x) q(x) will be true if both of them have the same set solution.

Example 11 :a. x + 2 if and only if 2x + 3 = x + 1b. If x 3 if and only if 4x 12c. ABC is a right triangle if and only if ABC have a right angle

D. PROBLEM SOLVING

1. Let p is a false proposition and q is a true proposition. Determine the truth value for each compound proposition below !a. p ~qb. ~p ~qc. ~(~p q)

d. ~(p q)e. p qf. ~(~p ~q)

Solution :a. …………………………b. …………………………c. …………………………

d. …………………………e. …………………………f. …………………………

2. Let p : ABC is a isosceles triangleq : ABC has the same 2 sides

Write the sentences from the following symbol !a. ~q pb. ~p ~qc. ~(~p q)

d. ~(p q)e. p qf. ~(p ~q)

Solution :a. …………………………b. …………………………c. …………………………

d. …………………………e. …………………………f. …………………………

3. Complete the tables below !

Logic Math .………………………………………………………………….. Hal : 19

p q p q ~p ~q ~( p q ) p ~qT T ……. ……. ……. …….T F ……. ……. ……. …….F T ……. ……. ……. …….F F ……. ……. ……. …….

4. Complete the following tables !

p q r (~p q) r ( p q ) ~r ~( p ~q ) rT T T ……. ……. …….T T F ……. ……. …….T F T ……. ……. …….T F F ……. ……. …….F T T ……. ……. …….F T F ……. ……. …….F F T ……. ……. …….F F F ……. ……. …….

E. EXERCISE 4

I. Complete the following blank !1. The truth value of “0 is a count number if and only if 0 is a nature number” is ….2. Let p is a false value and q is a true value. [(p ~q) (~p q)] have truth value

…………3. “ is irrational number if and only if 2log x + 2log 6 = 4”. The value of x in

order to make the proposition have TRUE value is ………..4. The truth value of “(sin - cos )2 = 1 + 2 sin cos if and only if

cot(270 + ) = tan ” is ………….5. “ x2 = 4 if and only if sin 150 = cos 300 ”. The value of x in order to make the

proposition have FALSE value is ………….

II. Choose the right answer of the following questions !1. The truth of ~p q is …………

a. F T T Fb. T F F F

c. T F T Fd. F T F T

e. F F F T

2. p q have the same value with…………..a. p ~qb. (p q) (q p)c. (q p) (~q p)

d. (q p)e. q p

3. The truth value of ~(p q) is ………..a. F F F Tb. F T T F

c. T F F Fd. T F F T

e. F T T T

4. The negation of p q is …………a. ~p ~qb. ~p q

c. ~p qd. ~p q

e. ~q ~p

5. Let p is false value, q is a true value and r is a true value. From the propositional below that have false value is ………….a. (p q) rb. (~p q) pc. p (q r)

d. ~p (q ~r)e. r (p q)

Logic Math .………………………………………………………………….. Hal : 20

6. The value of x in order to make the proposition : “3x + 12 9, x R if and only

if 5log 1 = ” have the TRUE value is ……..

a. x -1b. x -1

c. x 1d. x 1

e. x -1

7. The value of x in order to make the proposition : “ if and only if 5log

125 = 3” have the TRUE value is ………..a. -4b. 4

c. 3d. -3

e. 2

8. ~(~p q) will equivalent with ………….a. p qb. ~p ~q

c. p qd. ~p q

e. ~p q

III.Solve the problem below !1. Construct the truth table of (p q) ~q !

Solution : …………………………………………………………………………………………………………………………………………………………………………

2. Construct the truth table of (p q) (p ~q) !Solution : …………………………………………………………………………………………………………………………………………………………………………

3. Construct the truth table of (~p q) (r ~q) !Solution : …………………………………………………………………………………………………………………………………………………………………………

4. Prove that (p q) (~p q)Solution : …………………………………………………………………………………………………………………………………………………………………………

5. Prove that ~(p q) (p ~q) (~p q)Solution : …………………………………………………………………………………………………………………………………………………………………………

C. CONCEPT

TAUTOLOGI, CONTRADICTION AND CONTINGENCYTautology is a compound proposition that always have truth value. Contradiction is compound proposition that always have false value. Contingency is compound proposition that have truth and false values.

Example 12 : Show that [(p q) ~q] ~p is a Tautologi !

Solution : Look at the following truth tablep q ~p ~q p q (p q) ~q [(p q) ~q] ~pT T F F T F TT F F T F F TF T T F T F TF F T T T T T

So, based on the table, [(p q) ~q] ~p is a Tautologi

Example 13 : Show that (p q) (~p ~q) is a Contradiction !

Logic Math .………………………………………………………………….. Hal : 21

Solution : Look at the following truth tablep q ~p ~q p q ~p ~p (p q) (~p ~q)T T F F T F FT F F T T F FF T T F T F FF F T T F T F

So, based on the table, (p q) (~p ~q) is a Contradiction

NEGATION OF COMPOUND PROPOSITION1. De Morgan’s Duality Laws :

a. ~(p q) ~p ~qb. ~(p q) ~p ~q

2. ~(p q) p ~q3. ~(p q) ~p q p ~q (p ~q) (~p q)4. p q ~p q ~q ~p5. p q (p q) (q p)

Example 14 : Determine the negation of each propositions below !a. Semarang is the capitol city of Central Java and ATLAS cityb. 4 + 2 = 5 or 2 is a prime numberc. If all people pay tax so the development will be increased. 7 is an odd number if and only if 1 is a natural number

Solution :a. Semarang is not the capitol city of Central Java or not ATLAS cityb. 4 + 2 5 and 2 is not a prime numberc. All people pay tax and the development will not be increased. 7 is not an odd number if and only if 1 is a natural number

QUAINTERIZED PROPOSITIONSThere are 2 kinds of quainter :1. Existential Quainter

Quainter of “some ( several; there are / is )” is a statement which describes that several and every object should not have certain requirements. It symbolized with

and it read as follows “There is an x so that ……….”2. Universal Quainter

Quainter of “all” is a statement which describes that ever object fulfill certain requirements. It symbolized with and it read as follows “For all x or for every x”

Negation of q Quainterized Propositionsa. b.

Example 14 : Detemine the truth value for each quainter below !a.b.

Solution :

Logic Math .………………………………………………………………….. Hal : 22

a. is a false quainter because if the variable changed with 4, this open sentences will be false.

b. is a truth quainter because there is x = 6 will fulfill this open sentences become true.

CONVERSE, INVERSE AND CONTRAPOSITIONFrom Implication proposition “p q” can made a new propositionsConverse of “p q” : q pInverse of “p q” : ~p ~qContraposition of “p q” : ~q ~p

D. PROBLEM SOLVING

1. Write these quainters into logic symbol !a. For all odd number is divided by 1b. There is x as Real number so that 2x2 – 10x 0

Solution :a. ………………………………b. ………………………………

2. Determine the negation of the following quainters !a.b.

Solution :a. ………………………………b. ………………………………

3. Complete the following tables !p q ~p ~q p q ~p q ~(p q) (p ~q) (~p q)T T …. …. …. …. …. ….T F …. …. …. …. …. ….F T …. …. …. …. …. ….F F …. …. …. …. …. ….

4. Complete the following tables !p q r p q q r p r (p q) (q r) (p r)T T T …. …. …. ….T T F …. …. …. ….T F T …. …. …. ….T F F …. …. …. ….F T T …. …. …. ….F T F …. …. …. ….F F T …. …. …. ….F F F …. …. …. ….

5. Complete the following tables !p q r ~p ~r q r p ~r (p ~r) [(q r) (r ~p)]

Logic Math .………………………………………………………………….. Hal : 23

T T T …. …. …. ….T T F …. …. …. ….T F T …. …. …. ….T F F …. …. …. ….F T T …. …. …. ….F T F …. …. …. ….F F T …. …. …. ….F F F …. …. …. ….

E. EXERCISE 5

I. Complete the following blanks !1. The truth value of “ is ……………2. Negation of “There is x as Real number so that x2 + 10 = 8” is …………3. “All artist are singer”. The same value with these quainter is …………..4. “Some Quadratic Equation have Equal roots”. The same value these quainter

is ………….5. “If all prime number are odd number so 2 is not a prime number”. These

implication have inverse …………

II. Choose the right answer of the following questions !1. From the following compound proposition which have tautology is ………….

a. p (p ~p)b. p ~qc. ~q p

d. ~p (q p)e. ~p (q p)

2. From the following compound proposition which have contradiction is …………a. (p q) (~p q)b. ~p qc. ~p q

d. ~p qe. ~q q

3. “A goes to Jakarta or she doesn’t meet Susi”. It sentences has negation ……….a. A doesn’t go to Jakarta or meet Susib. A goes to Jakarta or meet Susic. A goes to Jakarta and meet Susid. A doesn’t go to Jakarta or doesn’t meet Susie. A doesn’t go to Jakarta and meet Susi

4. “If all students are graduated from High School so all teacher will be happy”. It sentences has negation …………...a. If all students do not graduated from High School so all teacher will not be

happyb. If one teacher is not happy so all students do not graduated from High Schoolc. If one student is not graduated from High School so all teacher will be sadd. All students do not graduated from High School so there are teacher will not

be happye. All students are graduated from High School so some teacher will not be

happy5. “If x is an even number so x is divided by 2” These sentences has the same value

with ………..a. If x is not divided by 2 so x is not an even numberb. If x is divided by 2 so x is not an even numberc. If x is not an even number so x is not divided by 2d. If x is divided by 2 so the x an even number

Logic Math .………………………………………………………………….. Hal : 24

e. x is not an even number so x is divided by 26. Negation of ~p ~q is ………….

a. ~p ~qb. p q

c. q pd. q p

e. ~(q p)

7. “If 5 + 6 11 so 7 is a prime number”. Negation of this sentences is …………a. 5 + 6 11 and 7 is not a prime numberb. If 5 + 6 11 so 7 is a not prime numberc. 5 + 6 11 and 7 is not a prime numberd. If 5 + 6 = 11 so 7 is a prime numbere. 5 + 6 11 so 7 is not a prime number

8. Negation of (p q) r is ……………a. (~p ~q) rb. (~p ~q) ~rc. p q ~r

d. (~p ~q) re. (~p ~q) r

9. Converse of ~p ~q is ………..a. ~q ~pb. q pc. q ~p

d. ~p qe. p q

10. Converse of (p q) r is ………..a. r (p q)b. r (p q)c. (~p ~q) ~r

d. ~r (~p ~q)e. ~r (~p ~q)

11. Inverse of p ~q is ………..a. p qb. ~q pc. q ~p

d. ~p qe. ~p ~q

12. “If there are sugar so there are ants”. Inverse from this sentences is ………..a. If there are no ants so there are sugarb. If there are no sugar so there are no antsc. If there are no ants so there are no sugard. If there are sugar so there are no antse. If there are no sugar so there are ants

13. Contraposition of ~q p is …………..a. ~p qb. p ~qc. q p

d. ~q ~pe. ~p ~q

14. Contraposition of (p q) r is ………..a. r (~p ~q)b. r (p q)c. ~r ~(p q)

d. ~(p q) ~re. ~r ~(p q)

15. “If 5 + 9 = 13 so 13 is an odd number”. Contraposition of this sentences is …….a. If 5 + 9 13 so 13 is an even numberb. If 5 + 9 = 13 so 13 is an even numberc. If 13 is an odd number so 5 + 9 13d. If 13 is an even number so 5 + 9 = 13e. If 13 is an even number so 5 + 9 13

16. “For all real numbers have addition invers”. Negation of this sentences is ………a. Some unreal numbers have addition inversb. For all unreal numbers have addition inversc. Some real numbers have addition inversd. There are real numbers that do not have addition inverse. For all real numbers do not have addition invers

17. Let is an odd number. This proposition have truth value if x member of …a. Count numberb. Round number

c. Nature numberd. Prime number

Logic Math .………………………………………………………………….. Hal : 25

e. Real number18. Negation of is …………..

a.b.c.

d.e.

19. The following propositions have truth vlue for , except ………..a.b.c.

d.e.

20. The following compound propositions is tautology, except …………a. (p q) pb. q (p q)c. ~p p

d. (p p) qe. p (p q)

III.Solve the problem below !1. Determine Converse, Inverse and Contraposition from each sentences below !

a. If x 0 so x positifb. If 25 = 16 so 6 is a prime numberc. If some students are lazy so all teachers will be sadd. If x2 – 2x – 3 so e. If PQR is a right triangle so P and R have acute angleSolution : a. ……………………………..b. ……………………………..c. …………………………….d. …………………………….e. …………………………….

2. Determine the truth value for each quainters below if A = {1, 2, 3, 4 }a.b.c.d.e.f.Solution :a. ……………………………b. ……………………………c. ……………………………

d. ……………………………e. ……………………………f. ……………………………

3. Determine the truth value of Inverse from (q ~p) (p ~q) !Solution : ……………………………………………………………………………………………………………………………………………………..................

4. Explore that ~p q (~q p) (p q) !Solution : ………………………………………………………………………………………………………………………………………………………………..……………………………………………………………………………………..

5. Find the negation of the fo;;owing compound propositions !a.b.Solution :a. …………………………

Logic Math .………………………………………………………………….. Hal : 26

b. …………………………

C. CONCEPT

DRAWING CONCLUTIONA determine or know set of a single or a compound one is called premise. A single proposition a compound one which is derived from some premises is called a conclution. Conclution of one or more premises whose truth has been proven and one conclution which is derived from its premises is called an argument. An argument is called valid if it can be proven that the argument is Tautology for all values of the premises truth. A simple method to proven an argument valid or not is use a truth table.

There are 3 simple method to draw conclution :1. Modus Ponens

Premis 1 : p qPremis 2 : p

Conclution : q

2. Modus TollensPremis 1 : p qPremis 2 : ~q

Conclution : ~p

3. SylogismPremis 1 : p qPremis 2 : q r

Conclution : p r

D. PROBLEM SOLVING

1. Determine the argument for each propositions below !a. All students are absent

Arjuna is absent

b. If Aswan passes the examination then he will gets prizeIf Aswan will gets prize then her sister wants too

c. If Lia cries so her tears is outHer tears is not aut

Solution : a. ………………………..b. ………………………..c. ………………………..d. ………………………..

Logic Math .………………………………………………………………….. Hal : 27

2. Determine the argument for each proposition below !a. p q

~p

q

b. p qp ~q

~p r

c. p ~qr q

p ~r

d. q rq s

p s

E. EXERCISE 6

I. Complete the following blanks !1. If x is a nature number so xis a count number

x is not a count numberThe conclution of its is ……………………

2. If sin (-) = sin (360 - ) so sin (90 - ) = cos If sin (90 - ) = cos so sin 180 = 0The conclution of its is ……………………

3. ~q ~ppThe conclution of its is ……………………

4. “If A is sick, A will go to doctor” . This sentences have the same meaning with .........................

5. If x is divided by 20 so x is divided by 4If x is divided by 4 so x is divided by 2The proposition include in …………………………

II. Choose the right answer of the folloing questions !1. Drawing conclution which including Modus Ponens is ……….

a. p q~q

~pb. p q

q

pc. p q

p

qd. p q

q r

p r

Logic Math .………………………………………………………………….. Hal : 28

e. p q~p

q

2. If I study so I will pass the examI studyThe conclution is ……………..a. I failed the exam b. I pass the examc. I’m not study

d. I study and pass the exame. I study

3. If so 5 is a prime number5 is an even numberThe conclution is ………………..a.b.

c.d. 5 is a prime numbere. 5 is even number

4. If x = 4 so x2 = 16If x2 = 16 so 4 is an even numberThe conclution is ………………..a. x = 4b. If x = 4 so 4 is an even numberc. x2 = 16

d. If 4 is an even number so x = 4e. x2 16

5. If 3log 27 = 3 so 3 is an odd number3 is an even numberThe conclution is ………………..a. 3log 27 = 3b. 3 is an even numberc. 3log 27 3

d. 3log 9 = 3e. 3 is an odd number

6. Let Premis 1 : If A is a student so A is clever Premis 2 : A is lazyThe conclution is ………………………a. A is cleverb. A is stupidc. A is a student

d. A is housewifee. A is lazy

7. (1). ~p q (2). p ~q (3). p r~p p q r

From the proposition above that have valid is …………….a. 1, 2, and 3b. 1 and 3c. 1 and 2

d. just 2e. just 3

8. Let : If A is a economical person so he rich If he rich so he will be happy He will not happyThe conclution is …………………….a. A is poorb. A rich and not economicalc. A is a economical person

d. He is riche. He is extravagantly

III. Solve the problem below !1. Determine the conclution for each proposition below !

Logic Math .………………………………………………………………….. Hal : 29

a. If x is real number so

b. If f(-x) = f(x) so f(x) is even functionCos(-x) = cos x

c. If holiday so B goes picnicB does not go picnic

d. If C is an artist so she is beautifulC is beautiful

Solution :a. …………………………………b. …………………………………c. …………………………………d. …………………………………

2. Use the truth table to cheque valid or in valid for the argument !a. p q

q

p

b. p qs r

(~p ~r) (q s)

Solution :a. …………………………………….

…………………………………….…………………………………….…………………………………….…………………………………….…………………………………….

b. …………………………………….…………………………………….…………………………………….…………………………………….…………………………………….…………………………………….

C. CONCEPT

PROOF IN MATH

INDIRECT PROOFSProving in mathematics that studied so far is direct proving ( include Modus Ponens, Modus Tollens and Sylogism ), meaning that proving a truth table is done by noticing that the truth is the effect of another statement that has been accepted as a truth of proven argument.

Logic Math .………………………………………………………………….. Hal : 30

PRINCIPAL OF MATHEMATICAL INDUCTIONLet : 1 + 3 + 5 + 7 + … If it want to make the easier calculation is :

S1 = 1 = 12 = 1S2 = 1 + 3 = 22 = 4S3 = 1 + 3 + 5 = 32 = 9 ………….. etcGeneraly, we get a conclution that Sn = n2, so 1 + 3 + 5 + 7 + … + (2n – 1) = n2.

To know it true or not, use the Principal of Mathemtics Induction; The truth of an infinitive sequence of proposition or a formula.

P(n) has the following characteristic(1). True for n = 1(2). True for n = k and True for n = k + 1The formula is true for

D. PROBLEM SOLVING

Using Mathematics Induction, Prove that 3 + 5 + 7 + … + (2n + 1) = n2 + 2nSolution :Let P(n) : 3 + 5+ 7 + … + (2n + 1) = n2 + 2nFor n = 1 2.1 + 1 = 12 + 2.1

2 + 1 = 1 + 23 = 3P(1) is true

Let n = k 3 + 5 + 7 + … + (2k + 1) = k2 + 2k is trueThen prove for n = k + 1 is also have truth valuen = k + 1 3 + 5 + 7 + … + (2k + 1) + [2(k+1)+1]

k2 + 2k + 2k + 3 k2 + 2k + 1 + 2k + 2 (k + 1)2 + 2(k + 1)

P(k + 1) is trueHence, P(n) is true

E. EXERCISE 7

I. Solve the problem below !1. Prove that “If n2 is an odd number so n is an odd number” using indirect proof !

Solution :

Logic Math .………………………………………………………………….. Hal : 31

…………………………………………………………………………………………………………………………………………………………………………

2. Prove that “If n2 is an even number so n is an even number” using contraposition proof !Solution :…………………………………………………………………………………………………………………………………………………………………………

3. Using Mathematics Induction, Prove that : a. 2 + 4 + 6 + 8 + … + 2n = n(n + 1)

b. 2 + 5 + 8 + 11 + … + (3n – 1) =

c. 4 + 9 + 14 + 19 + … + (5n-1) =

d. 1 + 3 + 6 + … +

e.

f.

Solution :a. ………………………………………….b. ………………………………………….c. ………………………………………….d. ………………………………………….e. ………………………………………….f. ………………………………………….

4. Using Mathematics Induction, Prove that :a. 1 + 2 + 22 + 23 + … + 2n-1 = 2n – 1

b. 3 + 32 + 33 + … + 3n =

Solution :a. ………………………………………….b. ………………………………………….

5. Using mathematics Induction, Prove that : a. 4n+1 – 4 is divided by 12b. 52n+1 + 1 is divided by 6c. 32n + 22n+2 is divided by 5d. n3 – n is divided by 24, for n members of odd number

Solution : a. …………………………………..b. …………………………………..c. …………………………………..d. …………………………………..

Logic Math .………………………………………………………………….. Hal : 32

The End