LOGI C MATHEMATIC MATERIAL by sumiyati trias

Logic Math .………………………………………………………………….. Hal : 1 CHAPTER V

LOGIC MATH A. GOALS FOR UNIT V 1. I will know the definition of sentences of proposition, open sentence and sentences of non-proposition. 2. I will know the symbol of propositional logic 3. I will be able to determine the negation of proposition 4. I will be able to determine the value and the truth table of conjunction 5. I will be able to determine the value and the truth table of disjunction 6. I will be able to apllicate conjunction and disjunction in circuit diagram 7. I will be able to determine the value and the truth table of implication 8. I will be able to determine the value and the truth table of bi implication 9. I will be to use De Morgan’s Duality Law 10. I will be able to determine the negation of implication 11. I will be able to determine the negation of bi implication 12. I will know the definition of tautology, contradiction and contingency 13. I will know the definition of Quanterized Proposition 14. I will be able to write the symbol of Universal Quainter 15. I will be able to determine the negation of Universal Quainter 16. I will be able to write the symbol of Existential Quainter 17. I will be able to determine the negation of Existential Quainter 18. I will be able to determine Converse, Inverse and Contraposition from Implies proposition 19. I will be able to draw conclution using direct proofs 20. I will know the step to proof math using indirect proofs 21. I will know the step to proof math using the Principal of Mathematical Induction 22. I will able to proof math using mathematical Induction B. WORDS TO REMEMBER Proposition Open sentences Set solution Circuit diagram Negation

Switching network Serial circuit Parallel circuit Antecedent Consequence

Conclution Premise Quainter Argument Valid

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 Sentences are sentences which contain variables and will be propositions if the variables are changed into constants in their universal sets Example 1 : • x – 5 > 23 • x2 – 2x + 1 = 0 • 4x + 5 = 40 3. Non Proposition A kind of a sentence which haven’t a true or false value

Logic Math .………………………………………………………………….. Hal : 2 Example 2 : a. 2 is a prime number b. Please, have a sit ! c. Mount Jayawijaya is in Aceh d. 3x2 + 8x + 5 < 0

( Proposition ) ( Not Proposition ) ( Proposition ) ( Open sentence )

• THE SYMBOL OF PROPOSITION LOGIC The symbols of certain propositions usually use “proposition variables” , like p, q or r a. Single Proposition p : I live in 15 Blimbing Street q : I study in State Junior High School b. Compound Proposition I study in State Junior High School AND live in 15 Blimbing Street My mother buys fruits OR vegetables There are 5 proposition operators and it can observe the table below NO 1. 2. 3. 4. 5.

OPERATORS NAME SYMBOL Negation ~ Conjunction ∧ Disjunction ∨ Implication ⇒ Bi implication ⇔

DAILY LANGUAGE No, not And, but, although, nevertheless Or If ……. so ……. If and only if …… so ….. if

• NEGATION OF A PROPOSITION If p is a proposition, then the negation of it, is put in notation as ~p or p . 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 values p ~p T F F T Note : T is True F is False

Logic Math .………………………………………………………………….. Hal : 3 D. PROBLEM SOLVING 1. State the following sentences proposition, not proposition or open sentences ! a. 23 is a rational number e. what is your father ? b. 6x – 3 = 9 f. Let’s go ! c. 63 is divided by 8 g. 3x – 9 > 18 d. 39 is an even number Solution : a. ………………….. e. ………………….. b. ………………….. f. ………………….. c. ………………….. g. ………………….. d. ………………….. 2. Determine the set solution from the problem below ! a. 2x – 1 < 5 , x ∈ N Solution : x = 1 ⇒ 2.1 – 1 < 5 ⇒ 1 < 5 x = 2 ⇒ ………… ⇒ …………. x = 3 ⇒ ………… ⇒ …………. x = 4 ⇒ ………… ⇒ …………. ∴ SS = { ………………… } b. x2 – 6x + 5 = 0, x ∈ N Solution : ……………………………………… ……………………………………… ……………………………………… ……………………………………… 3. Determine the negation of the following propositions and determine the truth values ! a. Palembang is the capital city of South Sulawesi b. sin 210° has positive value c. There are 12 month in a year d. Six is not a count number e. x2 – 6 ≥ 9 Solution : a. ……………………… d. ……………………… b. ……………………… e. ……………………… c. ……………………… 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 “ 625 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 delicious c. Taj Mahal is in Agra city b. 11 + 7 > 20 d. 5 + 2 < 0

Logic Math .………………………………………………………………….. Hal : 4 e. Borobudur temple is in Solo 2. From sentences below that an open sentences is ……….. a. 2 + 5 = 8 d. 2x – 2 > 10 b. 13 + 3 ≠ 13 e. Michael Jackson is a farmer c. The rose is blooming 3. From proposition below that have false value is ………… a. 2 is a prime number c. 2n is an even number d. The Sun rises in East b. 3 + 7 > 9 e. Zambia is in Africa 4. The negation of “All farmers are men” is ………….. a. All farmers are not women d. All farmers are women b. There is a woman farmer e. There is a man farmer c. Nor all farmers are women 5. The set solution from x2 – 6x + 5 ≥ 0 is …………. a. { x | x ≠ 5 or x ≠ 6 } d. { x | x ≤ -5 or x ≥ 6 } b. { x | x ≠ 1 or x ≠ 5 } e. { x | -5 ≤ x ≤ 1 } c. { x | x < -5 or x > 1 } 6. Negation of “3 is a count number” is …………… a. It’s not true that 3 is a count number b. 3 less than 5 c. 2 is not a count number d. 3 is not a prime number e. 3 is not a rational number 7. Negation of “There is x ∈ N so that x > 7” is ………….. a. For all x ∈ N, x > 7 d. For all x ∈ N, x ≥ 7 b. For all x ∈ N, x <7 e. There is x ∈ N, x ≤ 7 c. For all x ∈ N, x ≤ 7 8. Negation of “ 5x2 + 8x + 3 < 0 is ……………. 5 5 a. { x | x ≠ 1 or x ≠ 3 } d. { x | - 3 ≤ x ≤ -1 } b. { x | x ≤ -1 or x ≥

5 3

}

e. { x | -1 < x <

5 3

}

5 3

c. { x | x ≤ - or x ≥ -1 } 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 : …………………………………………………………………………………….. .…………………………………………………………………………………….

Logic Math .………………………………………………………………….. Hal : 5

C. CONCEPT • CONJUNCTIONS are 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 T T F F

q T F T F

p∧ q T F 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 = 5 b. 53 is prime number and all prime numbers are odd numbers Solution : 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 + 4 b. 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 true p : 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

Logic Math .………………………………………………………………….. Hal : 6 b. Let p : 51 is divided by 3 q : 3log x = -4 “p∧q” become true if both of them are true p : 51 is divided by 3 (T) 3 q : log x = -4 ( it must be true ) So, 3log x = -4 ⇔ 3-4 = x ⇔ x =

1 81

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

1 81

• DISJUNCTIONS are 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 T T F F

q T F T F

p∨ q F T T F

2. Inclusive Disjunctions A 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.

Logic Math .………………………………………………………………….. Hal : 7 The table of Inclusive Disjunctions : P T T F F

p∨ q T T T F

q T F T 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 = 7 b. 2 is a prime number or 5 − 2 = 3 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 =

18 or x2 – 5x – 6 = 0 9

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

•

APLICATION DIAGRAM

CONJUNCTION

AND

DISJUNCTION

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

q Figure 1.1

CIRCUIT

Logic Math .………………………………………………………………….. Hal : 8 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 p 1 1 0 0

p∧ q 1 0 0 0

q 1 0 1 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 1 1 0 0

q 1 0 1 0

p∨ q 1 1 1 0

D. PROBLEM SOLVING

1. Let p : 2 is a prime number q : 2 is an even number Write the logic symbol for each propositions below ! a. 2 is not a prime number and an even number b. 2 is a prime number and not an even number c. 2 is not a prime number and not an even number d. 2 is an even number or a prime number e. 2 is not an even number or not a prime number f. 2 is an even number or not a prime number

Logic Math .………………………………………………………………….. Hal : 9 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 ! a. ~p d. p ∨ ~q b. ~q e. ~p ∧ ~q c. ~p ∧ q f. ~p ∨ ~q Solution : a. …………………………. b. …………………………. c. ………………………….

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

3. Complete the following tables ! p

(p ∧ q)

T T T T F F F F

T T F F T T F F

T F T F T F T F

…. …. …. …. …. …. …. ….

( p ∧ q) ∨ r …. …. …. …. …. …. …. ….

(~p ∨ ~q) ∧ ~r …. …. …. …. …. …. …. ….

4. Complete the following tables ! p T T F F

q T F T F

~p …... …... …... …...

~q …... …... …... …...

~p ∧ q …... …... …... …...

p ∨ ~q …... …... …... …...

~p ∧ ~q …... …... …... …...

5. Determine the truth value from the following propositions ! a. 11 or 15 is divided by 3 b. sin2x + cos2x = 2 or 1 + tan2x = sec2x c. -5 < -1 or 8 is an old number d. 5log 125 = 3 and 33 = 26 e. Irrational numbers are surds and 16 = 4 f. (53)2 = 56 or 37.38 = 356 Solution : a. ………………………… b. ………………………… c. …………………………

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

Logic Math .………………………………………………………………….. Hal : 10 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 The symbol from proposition “Dean studies Physic or Math, but not studies Chemistry” is ………… 3. 2x2 + 10x + 8 < 0 and 31 is prime number The value of x in order to make the proposition nave TRUE value is …………. 4. A p q Symbol from circuit above is ……………...

5. 3x – 1 = 8 or sin 90° = 1 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 ∨ p) is ………… a. FFFT c. FFFF e. FTTF b. FFTF d. FTFF 2. ~(p ∨ ~q) have the same value with ………… a. ~p ∧ q c. p ∧ q e. ~p ∧ ~q b. ~p ∨ ~q d. p ∨ ~q 3. Negation of “There are people who don’t have sins” is ………….. a. Some people have sins d. All people have sins b. All people do not have sins e. People and sins c. No people have sins 4. Negation of “It’s not true that today is not rain” is ………….. a. Today is drizzle d. Today is rain b. Yesterday was hard rain e. Today is not rain yet c. Tomorrow will hard rain 5. The following proposition that include of conjunction, except …………. a. 242 = 576 and Indonesia is a Republic Nation b. Budi does not have a car although he is rich c. Although Iwan doesn’t study, he can graduates from school d. Ali is Teacher’s son but he’s veru stupid e. 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 = …………... 5 3 5 b. x < 3

a. x >

3 5 3 d. x > 5

c. x >

e. x ≥

5 3

7. The value of x in order “3log x = -4, x ∈ N or 7 < 5” have TRUE value is ……… 1 1 a. x = 81 c. x = 81 d. x ≠ 81 b. x ≠ -81 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 ) ∧ ~r d. ~( p ∨ ~q ) ∨ r b. ( p ∧ q ) ∧ r e. ( r ∧ q ) ∧ p c. ( q ∧ p ) ∨ ~r 9. The logic symbol of the following circuits is …………..

Logic Math .………………………………………………………………….. Hal : 11

p a. ( p ∨ q ) ∧ r b. ( p ∧ q ) ∨ r c. ( p ∨ q ) ∨ r

q r

d. ( ~p ∨ ~q ) ∧ r e. ( ~p ∧ q ) ∧ r

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

q T F T F

……. F T F F

a. ~p ∨ q b. ~p ∧ q

c. p ∧ ~q d. ~p ∧ ~q

e. ~p ∨ ~q

11. The truth value from the following table are ………… p T T T T F F F F

q T T F F T T F F

r T F T F T F T F

[( p ∧ ~q ) ∨ ~r] …… …… …… …… …… …… …… ……

a. F T T F T F T F b. F T T T F T F T c. T F F F T F T F

d. T F T F T F T F e. T T T T T T T F

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

q r

~q a. ( 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 ……….

Logic Math .………………………………………………………………….. Hal : 12 a. b. c. d. e.

1000 is divided by 6 or 12 is factor of 6 Frog is an amphibi animal and tiger is a wild animal Magelang is in Central Java although Madiun is in East Java 6 is an odd number and 2 is a prime number 2 x 3 = 7 or Surabaya is in East Java

14. Symbol of the truth table is ……… p T T F F

q T F T F

……. F F F F

a. ~p ∨ q b. p ∧ q

c. p ∨ ~q d. ~p ∧ ~q

e. ~p ∨ ~q

15. Symbol of the truth table is …………... p T T T T F F F F

q T T F F T T F F

r T F T F T F T F

………. F F F F T T 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 ! 40 = 2 10 and 2x – 5 ≥ 7 a. b. QE : x2 + 1 = 0 have equal roots or x −2 <3 x − 2 > 3 or 8 is a composit number c. 2 d. x - 4 > 0 and x2 – 2x – 3 < 0 Solution : a. ………………………………………. b. ………………………………………. c. ………………………………………. d. ………………………......................... 2. Let p : The Sun rises q : The day is shinny Write the sentences for each symbol below ! a. p ∧ ~q b. ~(p ∧ q) Solution : a. ……………………………………….

c. ~p ∧ ~q d. ~p ∧ q

Logic Math .………………………………………………………………….. Hal : 13 b. ………………………………………. c. ………………………………………. d. ………………………......................... 3. Construct the truth table for each propositions below ! a. ~p ∨ q d. (~r ∨ q ) ∧ ~r b. ~(p ∧ q) e. ~r ∨ ( p ∧ ~q) c. (p ∧ ~q) ∧ ~p f. ( ~p ∧ ~r) ∨ q 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.

1 = 2 + 1 and 1 is a prime factor of 21 2 −1

c. 61 is divided by 3 or log 40 – log 4 = log 20 Solution : a. ………………………………………. b. ………………………………………. c. ……………………………………….

C. CONCEPT • IMPLICATION Two 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; cause q is called conclution; consequence; effect The table of implication truth value : p T T F F

•

q T F T F

p⇒ q T F T T

LOGICAL IMPLICATION

Logic Math .………………………………………………………………….. Hal : 14 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 = 0 b. If x > 2 so x2 > 4 c. If PQRS is rectangle so ∠P = ∠Q = ∠R = ∠S = 90° d. If n is an odd number so ( 2n -1 ) is an odd number

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 ⇒ ~q d. ~ (p ⇒ q) b. ~p ⇒ ~q e. p ⇒ q c. ~ (~p ⇒ q) f. ~ (~p ⇒ ~q) Solution : a. …………………… d. b. …………………… e. c. …………………… f. 2. Let p : Juleha graduated from Junior High School q : Juleha will get a bicycle from her father Write the sentences from the following symbol ! a. ~q ⇒ p d. b. ~p ⇒ ~q e. c. ~ (~p ⇒ q) f.

…………………… …………………… ……………………

~ (p ⇒ q) p⇒q ~ (p ⇒ ~q)

Solution : a. ………………………… b. ………………………… c. ………………………… d. ………………………… e. ………………………… f. ………………………… 3. Complete the tables below ! p T T F F

q T F T F

p⇒ q …… …… …… ……

4. Complete the following tables !

~p ⇒ ~q …… …… …… ……

~ (p ⇒ q) …… …… …… ……

p ∧ ~q …… …… …… ……

Logic Math .………………………………………………………………….. Hal : 15 p

( p ∧ q) ⇒ r

( p ∨ q) ⇒ r

T T T T F F F F

T T F F T T F F

T F T F T F T F

…… …… …… …… …… …… …… ……

~ ( p ∨ q) ⇒ ~r …… …… …… …… …… …… …… ……

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 prize The 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 foods Symbol ~ ( p ⇒ q ) can be written as ………….. 1 5. “If 3 2 = 3 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 T c. F F F F e. T F T F b. T T T F d. F T T T 2. The truth value of ~ (p ⇒ q) is …………. a. F T F F c. F F T F e. T T T F b. T F T T d. T T F T 3. ~ (p ⇒ q) has the same value with ………… a. ~p ⇒ q c. p ∧ ~q e. q ∨ p b. p ⇒ ~q d. ~p ∧ q 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 ∧ q c. ~q ⇒ ~p e. ~p ∧ ~q b. ~p ∧ q d. q ⇒ p 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 late b. If all students are not come late so the day is not rain c. If the day is rain so all students will come late d. If the day is sunny so all students will not come late e. If some students come late so the day is rain

6. Negation of q ⇒ p is …………

Logic Math .………………………………………………………………….. Hal : 16 a. ~q ⇒ p c. ~q ⇒ p e. p ∧ ~q b. ~q ⇒ ~p d. ~p ∧ q 7. The value of x in order to make the proposition : “If x 2 – 3x = 0 so 7 is an even number” has a false value is ……….. a. 2 or 3 c. 1 or 3 e. 1 or -3 b. 3 d. 1 8. Symbol of the truth table is ………………. p q ……. T T F T F T F T T F F T a. p ⇒ ~q b. ~p ⇒ q

c. p ⇒ q d. q ⇒ p

e. ~q ⇒ ~p

9. Symbol of the truth table is ……………….. p q r ……….. T T T T T T F F T F T T T F F T F T T T F T F F F F T T F F F F a. (q ⇒ r) ⇒ p b. (p ⇒ r) ⇒ p

c. (p ⇒ q) ⇒ r d. p ⇒ (r ⇒ q)

e. r ⇒ (q ⇒ p)

10. Symbol of the truth table is ……………….. p q ……. T T T T F T F T T F F F a. (~p ∧ q) ⇒ ~q b. p ∧ q c. ~p ∨ q

d. ~q ⇒ ~p e. (~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

1 1 1 1 = 2 so = 2 2 3 3

c. If cos 90° = 1 so sin 270° = -1 d. If log 25 – log 5 = log 5 so 75 + e. If

1 −2 20 =

Solution :

6− 5

125 = 8 5

1 = −3 so 3log 27

Logic Math .………………………………………………………………….. Hal : 17 a. ……………………………. b. ……………………………. c. ……………………………. d. ……………………………. e. ……………………………. 2. Determine the value of x in order to make each propositions below have a TRUE value ! 1 = −1 so x3 – 1 = 0 3 1 x2 − 4 3 so ≤0 b. If tan 390° = 3 2x − 5

a. If 3log

c. If 1 – 3x ≥ 4 so 2 is a composit number 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) d. (r ∧ ~p) ⇒ q b. (p ∧ q) ⇒ ~r e. (~r ⇒ ~p) ∧ q c. (~r ∧ q) ⇒ ~p f. ( q ⇒ ~r) ∨ (p ⇒ ~q) Solution : a. ………………… d. ………………… b. ………………… e. ………………… c. ………………… f. ………………… 4. Construct the truth table for each propositions below ! a. q ⇒ p b. p ⇒ (r ⇒ q) c. (p ∧ q) ⇒ (p ∨ ~q) d. (p ⇒ q) ⇒ (~q ⇒ ~p) Solution : a. ………………….. c. …………………. b. ………………….. 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 = 0 b. If x2 – 16 = 0 so x – 4 = 0 c. If x2 < 4 so x < 2 d. If 2x2 + x – 1 = 0 so 2x – 1 = 0 e. If ∆ ABC is a isosceles triangle so AB = AC f. If PQRS is a rhombic so PQ = RS Solution : a. ………………. b. ………………. c. ……………….

d. ……………….. e. ………………. f. ……………….

C. CONCEPT

Logic Math .………………………………………………………………….. Hal : 18

• BI IMPLICATION Two 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 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 T T F F

q T F T F

p⇔ q T F F T

• LOGICAL BI IMPLICATION If 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 + 1 b. If x ≥ 3 if and only if 4x ≥ 12 c. ∆ 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 ⇔ ~q d. ~(p ⇔ q) b. ~p ⇔ ~q e. p ⇔ q c. ~(~p ⇔ q) f. ~(~p ⇔ ~q) Solution : a. ………………………… b. ………………………… c. …………………………

2. Let p : ∆ ABC is a isosceles triangle

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

Logic Math .………………………………………………………………….. Hal : 19 q : ∆ ABC has the same 2 sides Write the sentences from the following symbol ! a. ~q ⇔ p d. ~(p ⇔ q) b. ~p ⇔ ~q e. p ⇔ q c. ~(~p ⇔ q) f. ~(p ⇔ ~q) Solution : a. ………………………… b. ………………………… c. …………………………

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

3. Complete the tables below ! p T T F F

q T F T F

p⇔ q ……. ……. ……. …….

~p ⇔ ~q ……. ……. ……. …….

~( p ⇔ q ) ……. ……. ……. …….

p ⇔ ~q ……. ……. ……. …….

4. Complete the following tables ! p T T T T F F F F

q T T F F T T F F

r T F T F T F T F

(~p ∧ q) ⇔ r ……. ……. ……. ……. ……. ……. ……. …….

( p ⇔ q ) ⇒ ~r ……. ……. ……. ……. ……. ……. ……. …….

~( p ⇔ ~q ) ⇒ r ……. ……. ……. ……. ……. ……. ……. …….

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. “ 9 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 α - cos α)2 = 1 + 2 sin α cos α if and only if cot(270° + α ) = tan α” 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 F c. T F T F b. T F F F d. F T F T

2. p ⇔ q have the same value with………….. a. p ⇒ ~q

e. F F F T

b. (p ⇒ q) ∧ (q ⇒ p)

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

3. 4. 5.

c. (q ⇒ p) ∨ (~q ⇔ p) e. q ⇒ p d. (q ⇔ p) The truth value of ~(p ⇔ q) is ……….. a. F F F T c. T F F F e. F T T T b. F T T F d. T F F T The negation of p ⇔ q is ………… a. ~p ⇔ ~q c. ~p ⇒ q e. ~q ⇔ ~p b. ~p ∧ q d. ~p ⇔ q 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) ⇒ r d. ~p ⇒ (q ∨ ~r) b. (~p ∧ q) ⇔ p e. r ⇔ (p ∨ q) c. p ⇔ (q ∧ r) The value of x in order to make the proposition : “3x + 12 > 9, x ∈ R if and only if 5log 1 = a. x ≥ -1 b. x < -1

1 ” have the TRUE value is …….. 5

c. x > 1 d. x ≤ 1

e. x > -1 x

  7. The value of x in order to make the proposition : “   = 625 if and only if 1 5 

log 125 = 3” have the TRUE value is ……….. a. -4 c. 3 b. 4 d. -3 8. ~(~p ⇔ q) will equivalent with …………. a. p ⇒ q c. p ⇔ q b. ~p ⇒ ~q d. ~p ⇔ q

e. 2 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

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

• TAUTOLOGI, CONTRADICTION AND CONTINGENCY Tautology 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 table p q ~p ~q p⇒ q (p ⇒ q) ∧ ~q [(p ⇒ q) ∧ ~q] ⇒ ~p T T F F T F T T F F T F F T F T T F T F T F 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 ! Solution : Look at the following truth table p q ~p ~q p∨ q ~p ∧ ~p (p ∨ q) ⇔ (~p ∧ ~q) T T F F T F F T F F T T F F F T T F T F F F F T T F T F So, based on the table, (p ∨ q) ⇔ (~p ∧ ~q) is a Contradiction • NEGATION OF COMPOUND PROPOSITION 1. De Morgan’s Duality Laws : a. ~(p ∧ q) ≅ ~p ∨ ~q b. ~(p ∨ q) ≅ ~p ∧ ~q 2. ~(p ⇒ q) ≅ p ∧ ~q 3. ~(p ⇔ q) ≅ ~p ⇔ q ≅ p ⇔ ~q ≅ (p ∧ ~q) ∨ (~p ∧ q) 4. p ⇒ q ≅ ~p ∧ q ≅ ~q ⇒ ~p 5. 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 city b. 4 + 2 = 5 or 2 is a prime number c. If all people pay tax so the development will be increase d. 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 city b. 4 + 2 ≠ 5 and 2 is not a prime number c. All people pay tax and the development will not be increase d. 7 is not an odd number if and only if 1 is a natural number

• QUAINTERIZED PROPOSITIONS There are 2 kinds of quainter :

Logic Math .………………………………………………………………….. Hal : 22 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 ∃x 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 x and it read as follows “For all x or for every requirements. It symbolized with ∀ x” Negation of q Quainterized Propositions a. ~ [∀x, p ( x)] ≅ ∃x,[~ p ( x )] b. ~ [∃x, q ( x)] ≅ ∀x, [~ q ( x)] Example 14 : Detemine the truth value for each quainter below ! a. ∀x, x ∈R,2 x −1 = 3 b. ∃x, x ∈R,3 x −5 =13 Solution : a. ∀x, x ∈R,2 x −1 = 3 is a false quainter because if the variable changed with 4, this open sentences will be false. b. ∃x, x ∈R,3 x −5 =13 is a truth quainter because there is x = 6 will fulfill this open sentences become true. • CONVERSE, INVERSE AND CONTRAPOSITION From Implication proposition “p ⇒ q” can made a new propositions Converse of “p ⇒ q” : q⇒p Inverse of “p ⇒ q” : ~p ⇒ ~q Contraposition of “p ⇒ q” : ~q ⇒ ~p

D. PROBLEM SOLVING 1. Write these quainters into logic symbol ! a. For all odd number is divided by 1 b. There is x as Real number so that 2x2 – 10x < 0 Solution : a. ……………………………… b. ……………………………… 2. Determine the negation of the following quainters ! a. ∀x, x ∈ R, x 2 + 2 x + 4 < 0 b. ∃x, x ∈R, x ≥1 Solution : a. ……………………………… b. ………………………………

3. Complete the following tables ! p q ~p ~q p ∧ q ~p ⇔ q

~(p ⇔ q)

(p ⇒ ~q) ⇔ (~p ⇒ q)

Logic Math .………………………………………………………………….. Hal : 23 T T F F

T F T F

…. …. …. ….

4. Complete the following tables ! p q r p⇒ q q⇒ r T T T T F F F F

T T F F T T F F

T F T F T F T F

…. …. …. …. …. …. …. ….

5. Complete the following tables ! p q r ~p ∧ ~r 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 …. ….

…. …. …. …. p⇒ r …. …. …. …. …. …. …. ….

p ∨ ~r …. …. …. …. …. …. …. ….

…. …. …. …. (p ⇒ q) ∧ (q ⇒ r) ⇔ (p ⇒ r) …. …. …. …. …. …. …. ….

(p ∨ ~r) ⇒ [(q ⇒ r) ⇔ (r ⇒ ~p)] …. …. …. …. …. …. …. ….

E. EXERCISE 5 I. Complete the following blanks ! 1. The truth value of “ ∃ x, x ∈

R,2 x − 7 = x + 2" 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) d. ~p ∨ (q ∧ p) b. p ∨ ~q e. ~p ⇒ (q ∨ p) c. ~q ⇒ p 2. From the following compound proposition which have contradiction is ………… a. (p ∨ q) ⇔ (~p ∧ q) d. ~p ∧ q b. ~p ⇒ q e. ~q ∨ q c. ~p ∨ q 3. “A goes to Jakarta or she doesn’t meet Susi”. It sentences has negation ……….

Logic Math .………………………………………………………………….. Hal : 24 a. A doesn’t go to Jakarta or meet Susi b. A goes to Jakarta or meet Susi c. A goes to Jakarta and meet Susi d. A doesn’t go to Jakarta or doesn’t meet Susi e. 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 happy b. If one teacher is not happy so all students do not graduated from High School c. If one student is not graduated from High School so all teacher will be sad d. All students do not graduated from High School so there are teacher will not be happy e. All students are graduated from High School so some teacher will not be happy 5. “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 number b. If x is divided by 2 so x is not an even number c. If x is not an even number so x is not divided by 2 d. If x is divided by 2 so the x an even number e. x is not an even number so x is divided by 2 6. Negation of ~p ∧ ~q is …………. a. ~p ∨ ~q c. q ∨ p e. ~(q ∨ p) b. p ∧ q d. 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 number b. If 5 + 6 > 11 so 7 is a not prime number c. 5 + 6 ≤ 11 and 7 is not a prime number d. If 5 + 6 = 11 so 7 is a prime number e. 5 + 6 < 11 so 7 is not a prime number 8. Negation of (p ∧ q) ⇒ r is …………… a. (~p ∨ ~q) ∨ r d. (~p ∨ ~q) ∧ r b. (~p∧ ~q) ∨ ~r e. (~p ∧ ~q) ∧ r c. p ∧ q ∧ ~r 9. Converse of ~p ⇒ ~q is ……….. a. ~q ⇒ ~p d. ~p ⇒ q b. q ⇒ p e. p ⇒ q c. q ⇒ ~p 10. Converse of (p ∧ q) ⇒ r is ……….. a. r ⇒ (p ∨ q) d. ~r ⇒ (~p ∧ ~q) b. r ⇒ (p ∧ q) e. ~r ⇒ (~p ∨ ~q) c. (~p ∧ ~q) ⇒ ~r 11. Inverse of p ⇒ ~q is ……….. a. p ⇒ q d. ~p ⇒ q b. ~q ⇒ p e. ~p ⇒ ~q c. q ⇒ ~p 12. “If there are sugar so there are ants”. Inverse from this sentences is ……….. a. If there are no ants so there are sugar b. If there are no sugar so there are no ants c. If there are no ants so there are no sugar d. If there are sugar so there are no ants e. If there are no sugar so there are ants 13. Contraposition of ~q ⇒ p is …………..

Logic Math .………………………………………………………………….. Hal : 25 a. ~p ⇒ q d. ~q ⇒ ~p b. p ⇒ ~q e. ~p ⇒ ~q c. q ⇒ p 14. Contraposition of (p ∧ q) ⇒ r is ……….. a. r ⇒ (~p ∧ ~q) d. ~(p ∧ q) ⇒ ~r b. r ⇒ (p ∧ q) e. ~r ⇒ ~(p ∧ q) c. ~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 number b. If 5 + 9 = 13 so 13 is an even number c. If 13 is an odd number so 5 + 9 ≠ 13 d. If 13 is an even number so 5 + 9 = 13 e. 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 invers b. For all unreal numbers have addition invers c. Some real numbers have addition invers d. There are real numbers that do not have addition invers e. For all real numbers do not have addition invers 17. Let ∃x,2 x is an odd number. This proposition have truth value if x member of … a. Count number d. Prime number b. Round number e. Real number c. Nature number 18. Negation of ∀x, x ∈R,4 − x = 9 is ………….. a. ∃x, x ∈R,4 − x = 9 d. ∀x, x ∈R,9 − x = 4 b. ∀x, x ∈R,4 − x ≠ 9 e. ∃x, x ∈R,9 − x = 4 c. ∃x, x ∈R,4 − x ≠ 9 19. The following propositions have truth vlue for x ∈ R , except ……….. d. ∀x, x ∈ R,−(3 x +1) ≤ 0 a. ∀x, x ∈ R,5 x 2 − 7 > 0 e. ∀x, x ∈ R, x 2 + 6 < 0 b. ∀x, x ∈R, 3 +7 x ≥0 c. ∀x, x ∈R,3 x −4 ≥ −4 20. The following compound propositions is tautology, except ………… a. (p ∨ q) ∧ p d. (p ∧ p) ⇒ q b. q ⇒ (p ∨ q) e. p ⇒ (p ∨ q) c. ~p ∧ p III.Solve the problem below ! 1. Determine Converse, Inverse and Contraposition from each sentences below ! a. If x > 0 so x positif b. If 25 = 16 so 6 is a prime number c. If some students are lazy so all teachers will be sad d. If x2 – 2x – 3 ≤ 0 so − 1 ≤ x ≤ 3 e. If ∆ PQR is a right triangle so P and R have acute angle Solution : a. …………………………….. b. …………………………….. c. ……………………………. d. ……………………………. e. …………………………….

2. Determine the truth value for each quainters below if A = {1, 2, 3, 4 } a. ∃x, x ∈ A, x +1 = 6

Logic Math .………………………………………………………………….. Hal : 26 b. ∃x, x ∈ A,3 < 2 x −1 < 7 c. ∃x, x ∈ A,3 x − 6 < 2 x − 4 d. ∀x, x ∈ A, x + 2 < 6 e. ∀x, x ∈A, x −3 <4 f. ∀x, x ∈ A, x 2 − 3x + 2 = 0 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. ∀x, x ∈ A,2 x −12 > 0 b. ∀x, x ∈ A,3 x 2 − 2 x − 5 < 0 Solution : a. ………………………… b. …………………………

C. CONCEPT • DRAWING CONCLUTION A 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⇒q Premis 2 :p Conclution 2. Modus Tollens Premis 1 Premis 2 Conclution 3. Sylogism Premis 1 Premis 2

:q :p⇒q : ~q : ~p :p⇒q :q⇒r

Logic Math .………………………………………………………………….. Hal : 27 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 prize If Aswan will gets prize then her sister wants too c. If Lia cries so her tears is out Her tears is not aut Solution : a. ……………………….. b. ……………………….. c. ……………………….. d. ……………………….. 2. Determine the argument for each proposition below ! a. p ⇒ q c. p ⇒ ~q ~p r⇔q ∴q

∴ p ⇒ ~r

b. p ⇒ q p ⇒ ~q

d. q ∧ r q⇒s

∴ ~p ∧ r

∴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 number The conclution of its is …………………… 2. If sin (-α) = sin (360° - α) so sin (90° - α) = cos α If sin (90° - α) = cos α so sin 180° = 0 The conclution of its is …………………… 3. ~q ⇒ ~p p The conclution of its is ……………………

Logic Math .………………………………………………………………….. Hal : 28 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 4 If x is divided by 4 so x is divided by 2 The proposition include in ………………………… II. Choose the right answer of the folloing questions ! 1. Drawing conclution which including Modus Ponens is ………. a. p ⇒ q ~q ∴ ~p b. p ⇒ q q ∴p c. p ⇒ q p ∴q d. p ⇒ q q⇒r ∴p⇒r e. p ⇒ q ~p ∴q 2. If I study so I will pass the exam I study The conclution is …………….. a. I failed the exam b. I pass the exam c. I’m not study 3. If 25 = 5 so 5 is a prime number 5 is an even number The conclution is ……………….. 25 ≠ 5 a. 25 = −5 b. 4. If x = 4 so x2 = 16 If x2 = 16 so 4 is an even number The conclution is ……………….. a. x = 4 b. If x = 4 so 4 is an even number 2

c. x = 16 5. If 3log 27 = 3 so 3 is an odd number 3 is an even number The conclution is ……………….. a. 3log 27 = 3

d. I study and pass the exam e. I study

25 = 5 c. d. 5 is a prime number e. 5 is even number

d. If 4 is an even number so x = 4

e. x2 ≠ 16

b. 3 is an even number

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

log 27 ≠ 3

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

6. Let Premis 1 : If A is a student so A is clever Premis 2 : A is lazy The conclution is ……………………… a. A is clever d. A is housewife b. A is stupid e. A is lazy c. A is a student 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 3 d. just 2 b. 1 and 3 e. just 3 c. 1 and 2 8. Let : If A is a economical person so he rich If he rich so he will be happy He will not happy The conclution is ……………………. a. A is poor b. A rich and not economical c. A is a economical person

d. He is rich e. He is extravagantly

III. Solve the problem below ! 1. Determine the conclution for each proposition below ! a. If x is real number so x ≥0 x <0

b. If f(-x) = f(x) so f(x) is even function Cos(-x) = cos x c. If holiday so B goes picnic B does not go picnic d. If C is an artist so she is beautiful C 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 ⇒ q s⇒r ∴ (~p ∨ ~r) ∨ (q ⇒ s)

Logic Math .………………………………………………………………….. Hal : 30 Solution : a. ……………………………………. ……………………………………. ……………………………………. ……………………………………. ……………………………………. ……………………………………. b. ……………………………………. ……………………………………. ……………………………………. ……………………………………. ……………………………………. …………………………………….

C. CONCEPT •

PROOF IN MATH

INDIRECT PROOFS Proving 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. PRINCIPAL OF MATHEMATICAL INDUCTION Let : 1 + 3 + 5 + 7 + … If it want to make the easier calculation is : S1 =1 = 12 = 1 S2 =1+3 = 22 = 4 S3 =1+3+5 = 32 = 9 ………….. etc Generaly, 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 + 1 The formula is true for ∀n ∈ N

D. PROBLEM SOLVING Using Mathematics Induction, Prove that 3 + 5 + 7 + … + (2n + 1) = n2 + 2n

Logic Math .………………………………………………………………….. Hal : 31 Solution : Let P(n) : 3 + 5+ 7 + … + (2n + 1) = n2 + 2n For n = 1 → 2.1 + 1 = 12 + 2.1 2+1 =1+2 3 =3 ∴P(1) is true Let n = k → 3 + 5 + 7 + … + (2k + 1) = k2 + 2k is true Then prove for n = k + 1 is also have truth value n = 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 true Hence, P(n) is true ∀n ∈ N

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 : …………………………………………………………………………………… …………………………………………………………………………………… 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) 1 n( 3n +1) 2 1 c. 4 + 9 + 14 + 19 + … + (5n-1) = n( 5n + 3) 2 1 n( n + 1)( n + 2 ) d. 1 + 3 + 6 + … + n( n + 1) = 2 6 1 1 1 1 n + + + ... + = e. 1.3 3.5 5.7 ( 2n −1)( 2n + 1) 2n + 1 1 1 1 1 n + + + ... + = f. 1.2 2.3 3.4 n( n + 1) n + 1

b. 2 + 5 + 8 + 11 + … + (3n – 1) =

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

Logic Math .………………………………………………………………….. Hal : 32 f. …………………………………………. 4. Using Mathematics Induction, Prove that : a. 1 + 2 + 22 + 23 + … + 2n-1 = 2n – 1 b. 3 + 32 + 33 + … + 3n =

3 n (3 −1) 2

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

The

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