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1. Mathematical Logic

2. Mathematical Logic
The Statement
Open Sentences
Truth Value
Special Characteristic
Proposition
Not Proposition
Sentences
Explain
True
False
Variable : Variabel
Change
Manipulate
Certain Value
Constanta
Value
The Solution
Declare
Natation
Negation
Substitute
No
Not : Bukan
Not true that
Compound Statement
Connectives
And ( ᴧ )
Or ( ᴠ )
If ( →)
Then (→)
If and only if (↔)
Conjuction
Disjunction
Implication
Biimplication plikasi

3. Antecedent
Value of False
Equivalence
T
autology
Contradiction
Contingency
Consequent
Converse
Inverse
Contraposition
Conclusion
Premise
Syllogism
Commutative law
Associative law
Distributive law
De Morgan law

4. 1. Statement and open sentences
Definition
Proposition is a sentences, which can explain something true or false.
Definition
Open sentence is the sentence, not proposition. We can declare the open sentence as a
proposition by changing the variables with a certain value.

5. a.
b.
Answer :
a.
b.

6. 2. Notation and the Truth Value of a Proposition
lse.

7. 1.
2.
Answer :
1.
2.

8. 1.
Answer :
Because 𝜏 𝑝 = 𝑇, 𝑡ℎ𝑒𝑛 𝜏 ~𝑝 = 𝐹
p ~𝑝
T F
F T

9. A compound statement is a sentence that contains at least two simple proposition.
1. Conjunction
Definition
p q p ᴧ q
1.
Answer :
q : 2 is an even number
Based on the truth table of conjunction, because 𝜏 𝑝 = 𝑇 𝑎𝑛𝑑 𝜏 𝑞 = 𝑇, 𝑡ℎ𝑒𝑛 𝜏 𝑝ᴧ𝑞 = 𝐹.

10. 2. Disjunction
1.
Answer :
p q p ᴠ q

11. 3. Implication
1.
Answer :
q : log 15 – log 5 = log 3.
Based on truth table of implication, becauce 𝜏 (p) = F and 𝜏 (q) = T, thus 𝜏 (p⇒q) =
T.
p q p ⇒ q
T T T
T F F
F T T
F F T

12. 4. Biimplication
1.
Answer :
q : 24 = 16
Based on the truth table of biimplication, because 𝜏 (p) = T and 𝜏 (q) = T, thus 𝜏 (p↔q) =
T.
p q p↔q
T T T
T F F
F T F
F F T

13. 1. Equivalence
Definition
Two compound statements A and B are logically equivalent if they have the same truth
value, denoted by A ≡ B.
P q -p p ⇒ q -p v q
T T F T T
T F F F F
F T T T T
F F T T T

14. 1.
Answer :
~ 𝑝 ᷕ 𝑞 ≡ ~𝑝 ᷕ ~𝑞, and ~ 𝑝 ⇒
p q ~p ~q 𝑝 ᷕ
𝑞
𝑝 ᷕ 𝑞 𝑝 ⇒ 𝑞 ~ 𝑝 ᷕ
𝑞
~ 𝑝 ᷕ
𝑞
~ 𝑝 ⇒ 𝑞 ~𝑝 ᷕ ~
𝑞
~𝑝 ᷕ ~
𝑞
𝑝 ᷕ ~
𝑞
T T F F T T T F F F F F F
T F F T F T F T F T T F T
F T T F F T T T F F T F F
F F T T F F T T T F T T F

15. A. Commutative law :
1. 𝑝 ᷕ 𝑞 ≡ 𝑞 ᷕ 𝑝
2. 𝑝 ᷕ 𝑞 ≡ 𝑞 ᷕ 𝑝
B. Associative law
:
1.
2.
𝑝 ᷕ
𝑞
𝑝 ᷕ
𝑞
ᷕ 𝑟 ≡ 𝑝 ᷕ 𝑞 ᷕ
𝑟
ᷕ 𝑟 ≡ 𝑝 ᷕ 𝑞 ᷕ
𝑟
C. Distributive law :
1.
2.
𝑝 ᷕ 𝑞 ᷕ
𝑟
≡ 𝑝 ᷕ 𝑞 ᷕ 𝑝 ᷕ
𝑟
𝑝 ᷕ (𝑞ᷕ 𝑟) ≡ 𝑝 ᷕ 𝑞 ᷕ 𝑝 ᷕ
𝑟
D. De Morgan
law :
1. ~ 𝑝 ᷕ 𝑞 ≡ ~
𝑝 ᷕ ~𝑞
2. ~ 𝑝 ᷕ
𝑞 ≡ ~𝑝 ᷕ ~𝑞

16. 2. Tautology, Contradiction, and Contigency
a. Tautology
a

17. b. Contradiction

18. c. Contingency

19. • Converse is a statement, which has form of 𝑞 ⇒ 𝑝
• Inverse is a statement, which has form of ~𝑝 ⇒ ~𝑞
• Contraposition is a statement, which has form of ~𝑞 ⇒ ~
𝑝

20. 1.
2.
1.
Answers :
Contraposition : If it does not rain, then the sun will be shine.

21. 1. Modus Ponens
Premis 1 : p ⇒
q Premis 2 : p
Conclusion :
∴q
[( p ⇒ q) ᴧ p ] ⇒ q

22. 2. Modus Tollens
Premis 1 : p ⇒
q Premis 2 : ~q
Conclusion : ∴ ~p
[( p ⇒ q) ᴧ ~q ] ⇒
~p

23. li
3. Syllogism
Premis 1
Premis 2
: p ⇒
q
: q ⇒ r
Conclusion : ∴p ⇒
r

24. [( p ⇒ q) ᴧ (q ⇒ r)] ⇒ (p ⇒
r)