§1.2 Propositional Equivalence

1. §1.2 Propositional Equivalence • Two syntactically (i.e., textually) different compound propositions may be the semantically identical (i.e., have the same meaning). We call them equivalent. Learn: • Various equivalence rules or laws. • How to prove equivalences using symbolic derivations.

2. Tautologies and Contradictions A tautology is a compound proposition that is trueno matter what the truth values of its atomic (or component) propositions are! Ex.p  p[What is its truth table?] A contradictionis a compound proposition that is false no matter what! Ex.p  p[Truth table?] Other compound props. are contingencies.

3. Logical Equivalence p  q • Compound propositions p and q are logically equivalent to each other (written p q, or pq) IFFp and q contain the same truth values as each other in all rows of their truth tables. • Question: How many different propositions can be constructed from n propositional variables?

4. Proving Equivalence via Truth Tables Ex. Prove that pq  (p  q). F T T T F T T F F T T F T F T T F F F T

5. Proving Equivalence via Abbreviated Truth Tables Try to find a counter example Ex. Prove that pq  (p  q). Case 1: Try left side false, right side true • Assume p=F andq=F, then (p  q) =F. Case 2: Try right side false, left side true • Assume(p  q) F, then (p  q) T, then p  qT, thenp=F andq=F, then pq=F. We have exhausted all possibilities and not found a counterexample.

6. Logical Non-Equivalence Ex. p q and q p are not logically equivalent Prove that.

7. Equivalence Laws • Equivalence Laws 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. • Equivalent expressions can always be substituted for each other in a more complex expression - useful for simplification.

8. Equivalence Laws - 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 • Commutative: pq  qp pq  qp • Associative: (pq)r  p(qr) (pq)r  p(qr)

9. More Equivalence Laws • Distributive: p(qr)(pq)(pr)p(qr)(pq)(pr) • De Morgan’s:(p1p2…pn)  (p1p2…pn)(p1p2…pn)  (p1p2…pn) • Trivial tautology/contradiction:p  p  T p  p  F • Implication: pq  p q

10. More Equivalence Laws • Absurdity: (p q)  (p  q ) p • Contrapositive: pq  q  p • Absorption: p  (p  q)  p p  (p  q)  p • Exportation: (p  q)  r  p (q  R)

11. 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) p  q  (p  q)  ( p  q)

12. Tautology Example Demonstrate that [¬p(p q )]q is a tautology in two ways: • Using a truth table (did above) • Using a proof relying on “Equivalence Laws” to derive True through a series of logical equivalences

13. Tautology by proof [¬p(p q )]q  [(¬pp)(¬pq)]q Distributive  [ F  (¬pq)]q Trivial Contradiction  [¬pq ]q Identity ¬ [¬pq ] q Implies  [¬(¬p)¬q ] q DeMorgan  [p ¬q ] q Double Negation p  [¬q q ]Associative p  [q ¬q ]Commutative p  T Trivial Tautology  T Domination

14. Normal or Canonical Forms • Normal or Canonical Forms：Unique representations of a proposition • Examples: Construct a simple proposition of two variables which is true only when • P is true and Q is false: • P is true and Q is true: • P is true and Q is false or P is true and Q is true:

15. Disjunctive Normal Form A disjunction of conjunctions where • every variable or its negation is represented once in each conjunction (a minterm) • each minterms appears only once • Important in switching theory, simplification in the design of circuits.

16. To find the minterms of the DNF • Use the rows of the truth table where the proposition is 1 or True • If a zero appears under a variable, use the negation of the propositional variable in the minterm • If a one appears, use the propositional variable. • Example: Find the DNF of

17. Example • Find the DNF of • There are 5 cases where the proposition is true, hence 5 minterms. Rows 1,2,3, 5 and 7 produce the following disjunction of minterms:

18. Conjunctive Normal Form Similarly, Conjunctive Normal Form is a conjunction of disjunctions where • every variable or its negation is represented once in each disjunction (a maxterm) • each maxterms appears only once

19. To find the maxterms of the CNF • Use the rows of the truth table where the proposition is 0 or False • If a one appears under a variable, use the negation of the propositional variable in the maxterm • If a zero appears, use the propositional variable. • Example: Find the CNF of

20. Example • Find the CNF of • There are 3 cases where the proposition is false, hence 3 maxterms. Rows 4, 6 and 8 produce the following conjunction of maxterms: (P Q )¬R  (P ¬Q  ¬R) (¬P Q  ¬R) (¬P  ¬Q  ¬R)

21. Blackboard Exercises for 1.2 Worked out on the black-board. • “I don’t drink and drive” is logically equivalent to “If I drink, then I don’t drive”

22. Review: Propositional Logic (§§1.1-1.2) • Atomic propositions: p, q, r, … • Boolean operators:       • Compound propositions: s : (p q)  r • Equivalences:pq  (p  q) • Proving equivalences using: • Truth tables. • Symbolic derivations. p q  r … • Next: PREDICATE LOGIC