1. Propositional Equivalence A needed step towards proofs Copyright © 2014 Curt Hill

2. Preliminaries A tautology is a compound proposition that is always true –Regardless of the values of variables –p  p – law of excluded middle A contradiction is a compound proposition that is always false –p  p There are a number of theorems that are tautologies –They usually express an equivalence between two forms Copyright © 2014 Curt Hill

3. Equivalence Logical equivalence is when two variables or two compound propositions have the same truth value –Such as the same truth values in a truth table All tautologies are equivalent to each other –As are all contradictions Those are the trivial ones Copyright © 2014 Curt Hill

4. Example Tautology Copyright © 2014 Curt Hill pq p  q  q   pp  q   q  p TTTTT TFFFT FTTTT FFTTT

5. Logical Connective? Rosen states that the biconditional (  ) is a logical connective while the equivalence (  ) is only to be used when the biconditional yields a tautology Others use the two identically –Not my practice in Math 300, where I only use  as the connective Copyright © 2014 Curt Hill

6. Logical Equivalences The book gives several in tables in this section One of the purposes of these is in construction of proofs and other manipulation If an expression found on the right hand side of an equivalence is found in a expression, then the left hand side of the equivalence may be substituted –Effectively rewriting the expression Copyright © 2014 Curt Hill

7. Solving Problems In the Algebra of Real Numbers we do a similar thing We know that if we have this expression: x = 5(3 +2x) We may use the distributivity law x(y + z) = xy + xz to obtain x = 15+10x The logical equivalences are the laws of Boolean algebra Copyright © 2014 Curt Hill

8. Towards Proofs A truth table may prove (or disprove) equivalence However, as the number of variables increase the size of a truth table becomes unwieldy The alternative is to construct proofs by rewriting Copyright © 2014 Curt Hill

9. Form of a Proof Depends on the author Starts with a supposition –The desired theorem We rewrite the supposition and give the justification –The justification is a theorem, known to be valid We quit when we are left with T as the result Copyright © 2014 Curt Hill

10. Prove: (p  q)  p The supposition is (p  q)  p We next start a series of rewrites based on logical equivalences stated in table 6 or 7 of the book –Table 7 by line –Table 6 by name Sometimes we change the letters if they conflict with the law that we are citing –However, not in this one Copyright © 2014 Curt Hill

11. Prove: (p  q)  p (p  q)  p   (p  q)  p –Table 7, first line (p  q)  p  (  p   q)  p –deMorgan’s (p  q)  p  p  (  p   q) –Commutativity (p  q)  p  (p   p)   q –Associativity (p  q)  p  T   q –Negativity (p  q)  p  T –Dominatation Copyright © 2014 Curt Hill

12. Commentary The book shows this with the justification on the same line –Difficult for PowerPoint It also never rewrites the supposition – just leaves it blank Here the justification follows the rewrite, but could precede as well Copyright © 2014 Curt Hill

13. Satisfiability Consider three classes of logical expression based on their values in a truth table –Tautology – all true –  p  p –Contingency – mixed values – p  q –Contradiction – all false –  p  p The first two are satisfiable, with a proper choice of values they can be made true –Any choice that is true is a solution The third is unsatisfiable Copyright © 2014 Curt Hill

14. Showing Satisfiability Several ways Find a solution Show that the expression is a tautology Showing unsatisfiability is same as satisfiability of the negation Copyright © 2014 Curt Hill

15. Exercises Section 1.3 of Rosen Problems: 5, 7, 23(no truth table), 41, 61 Copyright © 2014 Curt Hill