1. The document discusses mathematical logic and its key concepts such as propositions, truth values, compound statements, logical equivalences, and logical rules.
2. It defines terms like tautology, contradiction, and contingency and explains logical connectives like conjunction, disjunction and implication.
3. The document also covers logical laws like commutative law, associative law and De Morgan's law as well as logical rules like modus ponens, modus tollens and syllogisms.
With vocabulary
1. The Statements, Open Sentences, and Trurth Values
2. Negation
3. Compound Statement
4. Equivalence, Tautology, Contradiction, and Contingency
5. Converse, Invers, and Contraposition
6. Making Conclusion
With vocabulary
1. The Statements, Open Sentences, and Trurth Values
2. Negation
3. Compound Statement
4. Equivalence, Tautology, Contradiction, and Contingency
5. Converse, Invers, and Contraposition
6. Making Conclusion
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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
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.
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.
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
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
