logic

1. LAWS OF ALGEBRA OF PREPOSITIONS
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Identities
 𝑝 ∨ 𝑝 ≡ 𝑝
 𝑝 ∨ 𝑇 ≡ 𝑇
 𝑝 ∨ 𝐹 ≡ 𝑝
 𝑝 ∧ 𝑝 ≡ 𝑝
 𝑝 ∧ 𝑇 ≡ 𝑝
 𝑝 ∧ 𝐹 ≡ 𝐹
 𝑝 → 𝑝 ≡ 𝑇
 𝑝 → 𝑇 ≡ 𝑇
 𝑝 → 𝐹 ≡∼ 𝑝
 𝑇 → 𝑝 ≡ 𝑝
 𝐹 → 𝑝 ≡ 𝑇
 𝑝 ↔ 𝑝 ≡ 𝑇
 𝑝 ↔ 𝑇 ≡ 𝑝
 𝑝 ↔ 𝐹 ≡∼ 𝑝

2. LAWS OF ALGEBRA OF PREPOSITIONS
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Commutative Law
 𝑝 ∨ 𝑞 ≡ 𝑞 ∨ 𝑝
 𝑝 ∧ 𝑞 ≡ 𝑞 ∧ 𝑝
 𝑝 → 𝑞 ≠ 𝑞 → 𝑝
 𝑝 ↔ 𝑞 ≡ 𝑞 ↔ 𝑝
The Law of Double
Negation
 ∼ ∼ 𝑝 ≡ 𝑝
Complement Law
 𝑝 ∨∼ 𝑝 ≡ 𝑇
 𝑝 ∧∼ 𝑝 ≡ 𝐹
 𝑝 →∼ 𝑝 ≡∼ 𝑝
 ∼ 𝑝 → 𝑝 ≡ 𝑝
 𝑝 ↔∼ 𝑝 ≡ 𝐹

3. LAWS OF ALGEBRA OF PREPOSITIONS
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Associative Law
 𝑝 ∨ 𝑞 ∨ 𝑟 ≡ 𝑝 ∨ 𝑞 ∨ 𝑟
 𝑝 ∧ 𝑞 ∧ 𝑟 ≡ 𝑝 ∧ 𝑞 ∧ 𝑟
Distributive Law
 𝑝 ∨ 𝑞 ∧ 𝑟 ≡ 𝑝 ∨ 𝑞 ∧ (𝑝 ∨ 𝑟)
 𝑝 ∧ 𝑞 ∨ 𝑟 ≡ 𝑝 ∧ 𝑞 ∨ (𝑝 ∧ 𝑟)

4. LAWS OF ALGEBRA OF PREPOSITIONS
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The Law of Absorption
 𝑝 ∨ 𝑝 ∧ 𝑞 ≡ 𝑝
 𝑝 ∧ 𝑝 ∨ 𝑞 ≡ 𝑝
De Morgan’s Law
 ∼ 𝑝 ∨ 𝑞 ≡∼ 𝑝 ∧∼ 𝑞
 ∼ 𝑝 ∧ 𝑞 ≡∼ 𝑝 ∨∼ 𝑞

5. LAWS OF ALGEBRA OF PREPOSITIONS
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The Law of Equivancy of Contrapositive
 𝑝 → 𝑞 ≡∼ 𝑞 →∼ 𝑝
The Law of Syllogism
 If 𝑝 → 𝑞 and 𝑞 → 𝑟 then 𝑝 → 𝑟
Other Laws
 𝑝 → 𝑞 ≡∼ 𝑝 ∨ 𝑞
 𝑝 ↔ 𝑞 ≡ 𝑝 → 𝑞 ∧ (𝑞 → 𝑝)

6. TAUTOLOGIES AND CONTRADICTIONS
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Tautology
 A compound statement that is always true no
matter what truth values are assigned to its
component propositions is called a tautology
 Examples: Consider the following truth tables
𝒑 ∼ 𝒑 (𝒑 ∨∼ 𝒑)
𝑇 𝐹 𝑻
𝐹 𝑇 𝑻

7. TAUTOLOGIES AND CONTRADICTIONS
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Tautology
 Examples: Consider the following truth tables
𝒑 𝒒 ∼ 𝒑 𝒑 ∨ 𝒒 ∼ 𝒑 ∨ (𝒑 ∨ 𝒒)
𝑇 𝑇 𝐹 𝑇 𝑻
𝑇 𝐹 𝐹 𝑇 𝑻
𝐹 𝑇 𝑇 𝑇 𝑻
𝐹 𝐹 𝑇 𝐹 𝑻

8. TAUTOLOGIES AND CONTRADICTIONS
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Contradictions/Fallacy
 These are compound statements which are always
false no matter what truth values are assigned to
the component propositions
 Examples: Consider the following truth tables
𝒑 ∼ 𝒑 (𝒑 ∧∼ 𝒑)
𝑇 𝐹 𝑭
𝐹 𝑇 𝑭

9. TAUTOLOGIES AND CONTRADICTIONS
Mayengo,
M.
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Contradictions
 Examples: Consider the following truth tables
𝒑 𝒒 𝒑 ∧ 𝒒 𝒑 ∨ 𝒒 ∼ (𝒑 ∨ 𝒒) 𝒑 ∧ 𝒒 ∧∼ (𝒑 ∨ 𝒒)
𝑇 𝑇 𝑇 𝑇 𝐹 𝑭
𝑇 𝐹 𝐹 𝑇 𝐹 𝑭
𝐹 𝑇 𝐹 𝑇 𝐹 𝑭
𝐹 𝐹 𝐹 𝐹 𝑇 𝑭

10. TAUTOLOGIES AND CONTRADICTIONS
Contingency
 These are compound statements which are neither
false nor true no matter what truth values are
assigned to the component propositions
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11. TAUTOLOGIES AND CONTRADICTIONS
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Logical Equivalence
 Two or more propositions are said to be logically equivalent
(or equal) if they have the same/identical truth values.
 Logical equivalence is denoted by symbol " = "or " ≡ “
 Examples: Consider the truth values of the propositions ∼ (𝑝 ∧
𝑞) and ∼ 𝑝 ∨∼ 𝑞 bellow.
𝒑 𝒒 ∼ 𝒑 ∼ 𝒒 𝒑 ∧ 𝒒 ∼ (𝒑 ∧ 𝒒) ∼ 𝒑 ∨∼ 𝒒
𝑇 𝑇 𝐹 𝐹 𝑇 𝑭 𝑭
𝑇 𝐹 𝐹 𝑇 𝐹 𝑻 𝑻
𝐹 𝑇 𝑇 𝐹 𝐹 𝑻 𝑻
𝐹 𝐹 𝑇 𝑇 𝐹 𝑻 𝑻

12. TAUTOLOGIES AND CONTRADICTIONS
Mayengo,
M.
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Logical Equivalence
 The propositions ∼ (𝑝 ∧ 𝑞) and ∼ 𝑝 ∨∼ 𝑞 bellow have the
same truth values for all possible ways of assigning truth
values to the component propositions 𝑝 and 𝑞.
 Hence, we conclude that ∼ 𝑝 ∧ 𝑞 =∼ 𝑝 ∨∼ 𝑞
𝒑 𝒒 ∼ 𝒑 ∼ 𝒒 𝒑 ∧ 𝒒 ∼ (𝒑 ∧ 𝒒) ∼ 𝒑 ∨∼ 𝒒
𝑇 𝑇 𝐹 𝐹 𝑇 𝑭 𝑭
𝑇 𝐹 𝐹 𝑇 𝐹 𝑻 𝑻
𝐹 𝑇 𝑇 𝐹 𝐹 𝑻 𝑻
𝐹 𝐹 𝑇 𝑇 𝐹 𝑻 𝑻

13. TAUTOLOGIES AND CONTRADICTIONS
Mayengo,
M.
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Logical Equivalence
 Similarly, we can show that 𝑝 → 𝑞 =∼ 𝑝 ∨ 𝑞.
 The truth table bellow proves the equivalence of the two
compound statements.
𝒑 𝒒 ∼ 𝒑 𝒑 → 𝒒 ∼ 𝒑 ∨ 𝒒
𝑇 𝑇 𝐹 𝑻 𝑻
𝑇 𝐹 𝐹 𝑭 𝑭
𝐹 𝑇 𝑇 𝑻 𝑻
𝐹 𝐹 𝑇 𝑻 𝑻

14. QUESTIONS
 By using truth table and laws of proposition logic,
classify the following mathematic logic formula
whether they are tautology, contradiction or
contingency statements.
 (i)(∼p→(p→q)
 (ii) (p→q)(q→p)
 (iii) p(p→q)
 (iv) (pq)→(rp)
 (v) (pq)(∼pr)
Mayengo,
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