Equivalence Formulas / Laws of Logic

Overview

• Discussing two approaches to check equivalence of propositions:
  1. Using truth tables
  2. Using equivalence formulas (laws of logic)

10 Laws of Logic

1. Idempotent Law

• Disjunction (OR):

  • p ∨ p is equivalent to p
  • If p is true: true ∨ true = true
  • If p is false: false ∨ false = false

• Conjunction (AND):

  • p ∧ p is equivalent to p
  • If p is true: true ∧ true = true
  • If p is false: false ∧ false = false

2. Identity Law

• Disjunction:
  • p ∨ false is equivalent to p
• Conjunction:
  • p ∧ true is equivalent to p

3. Dominance Law

• Disjunction:
  • p ∨ true is always true
• Conjunction:
  • p ∧ false is always false

4. Associative Law

• Disjunction:
  • (p ∨ q) ∨ r is equivalent to p ∨ (q ∨ r)
• Conjunction:
  • (p ∧ q) ∧ r is equivalent to p ∧ (q ∧ r)

5. Distributive Law

• Disjunction over Conjunction:
  • p ∨ (q ∧ r) is equivalent to (p ∨ q) ∧ (p ∨ r)
• Conjunction over Disjunction:
  • p ∧ (q ∨ r) is equivalent to (p ∧ q) ∨ (p ∧ r)

6. Commutative Law

• Disjunction:
  • p ∨ q is equivalent to q ∨ p
• Conjunction:
  • p ∧ q is equivalent to q ∧ p

7. Absorption Law

• First Rule:
  • p ∨ (p ∧ q) is equivalent to p
• Second Rule:
  • p ∧ (p ∨ q) is equivalent to p

8. De Morgan's Law

• First Rule:
  • ¬(p ∨ q) is equivalent to ¬p ∧ ¬q
• Second Rule:
  • ¬(p ∧ q) is equivalent to ¬p ∨ ¬q

9. Negation Law

• First Rule:
  • p ∨ ¬p is always true
• Second Rule:
  • p ∧ ¬p is always false

10. Double Negation Law

• ¬(¬p) is equivalent to p

Conclusion

• Using these laws, we can check the equivalence of two propositions without relying on truth tables.
• Future videos will cover examples using these approaches.