Week-4 Lecture-2_Logical_Implication_Equivalence_Laws (1)

CSEC1001: Foundations of Computing and Cyber Security

Week 4 - Lecture 2: Logical Implications and Equivalence Laws

1. Overview of Concepts

• Conditional Statements: Formulated as "If p then q", represented as p → q.

• Biconditional Statements: Formulated as "p if and only if q", denoted as p ↔ q.

2. Evaluating Truth Values

2.1 Truth Tables for p → q and p ↔ q

• For p → q:

  • T (true) → T (true) = T

  • T → F = F

  • F → T = T

  • F → F = T

• For p ↔ q:

  • T ↔ T = T

  • T ↔ F = F

  • F ↔ T = F

  • F ↔ F = T

3. Logical Connectives

• Negation (Not): Flips truth value.

• Conjunction (And): True if both p and q are true.

• Disjunction (Or): True if at least one of p or q is true.

• Implication (If... Then): False only if p is true and q is false.

• Biconditional (If and Only If): True if both p and q share the same truth value.

4. Implication Detailed Analysis

4.1 Key Properties

• If p is false, then p → q is automatically true regardless of q's truth value.

• p → q is logically equivalent to ¬p ∨ q.

5. Converse, Inverse, Contrapositive

5.1 Definitions

• Converse: q → p (hypothesis and conclusion are swapped).

• Inverse: ¬p → ¬q (negate both the hypothesis and conclusion).

• Contrapositive: ¬q → ¬p (swap and negate both).

5.2 Logical Equivalence

• The contrapositive of a conditional statement is logically equivalent to the original conditional.

6. Logical Equivalence Laws

6.1 Definition

• Two propositions are logically equivalent if they yield the same truth values across all possible scenarios.

6.2 Examples

• Example 1: ¬(p ∧ q) ↔ ¬p ∨ ¬q (De Morgan’s Law).

• Example 2: ¬(p ∨ q) ↔ ¬p ∧ ¬q.

7. Laws of the Algebra of Propositions

7.1 Key Laws

• Idempotent Laws: p ∧ p ↔ p; p ∨ p ↔ p.

• Commutative Laws: p ∧ q ↔ q ∧ p; p ∨ q ↔ q ∨ p.

• Associative Laws: p ∧ (q ∧ r) ↔ (p ∧ q) ∧ r; p ∨ (q ∨ r) ↔ (p ∨ q) ∨ r.

• Distributive Laws: p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r); p ∨ (q ∧ r) ↔ (p ∨ q) ∧ (p ∨ r).

• Complement Laws: p ∧ ¬p ↔ F; p ∨ ¬p ↔ T.

• Double Negation: ¬(¬p) ↔ p.

8. Summary of Logical Equivalence and Laws

• Understanding how to manipulate logical expressions using these principles is essential in computing and cybersecurity fields.

• These concepts assist in simplifying logical statements and developing more complex logical arguments.