CS 285- Discrete Mathematics

1. CS 285- Discrete Mathematics Lecture 3

2. Section 1.2 Propositional equivalences • Two syntactically (i.e. textually) different compound propositions may be semantically identical (i.e., have the same meaning). We call them equivalent. • Learning outcome: • Various equivalence rules or laws. • How to prove equivalences using symbolic derivations. Propositional equivalences

3. Tautologies and Contradictions • A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are! • Ex. p ∨ ¬p [What is its truth table?] • A contradiction is a compound proposition that is false no matter what! • Ex. p ∧ ¬p [Truth table?] • Other compound props. are contingencies. Propositional equivalences

4. Logical Equivalence • Compound proposition p is logically equivalent to compound proposition q, written p⇔ q, IFF the compound proposition p↔q is a tautology. • Compound propositions p and q are logically equivalent to each other IFF p and q contain the same truth values as each other in all rows of their truth tables. Propositional equivalences

5. Proving Equivalence via Truth Tables • Ex. Prove that p ∨ q ⇔ ¬(¬p ∧ ¬q) Propositional equivalences

6. Equivalence Laws • These are similar to the arithmetic identities you may have learned in algebra, but for propositional equivalences instead. • They provide a pattern or template that can be used to match all or part of a much more complicated proposition and to find an equivalence for it. • Examples: • Identity: p∧T ⇔ p p∨F ⇔ p • Domination: p∨T ⇔ T p∧F ⇔ F • Idempotent: p∨p ⇔ p p∧p ⇔ p • Double negation: ¬¬p ⇔ p Propositional equivalences

7. Equivalence Laws - Examples • Commutative: p∨q ⇔ q∨pp∧q ⇔ q∧p • Associative: (p∨q)∨r ⇔ p∨(q∨r) (p∧q)∧r ⇔ p∧(q∧r) • Distributive: p∨(q∧r) ⇔ (p∨q)∧(p∨r) p∧(q∨r) ⇔ (p∧q)∨(p∧r) • De Morgan’s: ¬(p∧q) ⇔ ¬p ∨ ¬q ¬(p∨q) ⇔ ¬p ∧ ¬q • Trivial tautology/contradiction: p ∨ ¬p ⇔ T p ∧ ¬p ⇔ F Propositional equivalences

8. Defining Operators via Equivalences • Using equivalences, we can define operators in terms of other operators • Exclusive or: p⊕q ⇔ (p∨q)∧¬(p∧q) p⊕q ⇔ (p∧¬q)∨(q∧¬p) • Implies: p→q ⇔ ¬p ∨ q • Biconditional: p↔q ⇔ (p→q) ∧ (q→p) p↔q ⇔ ¬(p⊕q) Propositional equivalences

9. Proving Equivalences • We can prove equivalences using: • Truth tables. • Symbolicderivations. p ⇔ q ⇔ r … Propositional equivalences

10. Prove Equivalences using a Truth Table • Examples: 1- Show that ¬ (p ∨ q) and ¬ p ∧ ¬ q are logically equivalent 2- Show that p → q and ¬ p ∨ q are logically equivalent 3- Show that p ∨(q ∧ r) and (p ∨ q) ∧(p ∨ r) are logically equivalent Propositional equivalences

11. Prove Equivalences using Symbolic Derivations -- I • Show that ¬ (p → q) and p ∧ ¬ q are equivalent • ¬ (p → q) ⇔ ¬(¬ p ∨ q) (shown earlier) ⇔ ¬(¬ p) ∧ ¬ q (De Morgan’s) ⇔ p ∧ ¬ q (Double Negation) Propositional equivalences

12. Prove Equivalences using Symbolic Derivations -- II • Show that (p ∧ q) → (p ∨ q) is a tautology • (p ∧q)→(p ∨ q) ⇔ ¬(p ∧q) ∨ (p ∨ q) ⇔ (¬p ∨ ¬q) ∨ (p ∨ q) ⇔ (¬p ∨ p) ∨ (¬q ∨ q) ⇔ T∨ T ⇔ T Propositional equivalences

13. Prove Equivalences using Symbolic Derivation -- III • Check using a symbolic derivation whether: (p ∧ ¬q) → (p ⊕ r) ⇔ ¬p ∨ q ∨ ¬r [Expand definition of →] (p ∧ ¬q) → (p ⊕ r) ⇔ ¬(p ∧ ¬q) ∨ (p ⊕ r) [Expand definition of ⊕] ⇔ ¬(p ∧ ¬q) ∨ ((p ∨ r) ∧ ¬(p ∧ r)) [DeMorgan’s Law] ⇔ (¬p ∨ q) ∨ ((p ∨ r) ∧ ¬(p ∧ r)) [∨ commutes] ⇔ (q ∨ ¬p) ∨ ((p ∨ r) ∧ ¬(p ∧ r)) [∨ associative] ⇔ q ∨ (¬p ∨ ((p ∨ r) ∧ ¬(p ∧ r))) [distrib. ∨ over ∧] ⇔ q ∨ (((¬p ∨ (p ∨ r)) ∧ (¬p ∨ ¬(p ∧ r))) Propositional equivalences

14. Cont. • [assoc.] ⇔ q ∨ (((¬p ∨ p) ∨ r) ∧ (¬p ∨ ¬(p ∧ r))) • [trivial taut.] ⇔ q ∨ ((T ∨ r) ∧ (¬p ∨ ¬(p ∧ r))) • [domination] ⇔ q ∨ (T ∧ (¬p ∨ ¬(p ∧ r))) • [identity] ⇔ q ∨ (¬p ∨ ¬(p ∧ r)) • [DeMorgan’s] ⇔ q ∨ (¬p ∨ (¬p ∨ ¬r)) • [Assoc.] ⇔ q ∨ ((¬p ∨ ¬p) ∨ ¬r) • [Idempotent] ⇔ q ∨ (¬p ∨ ¬r) • [Assoc.] ⇔ (q ∨ ¬p) ∨ ¬r • [Commut.] ⇔ ¬p ∨ q ∨ ¬r Propositional equivalences