Proofs Using Logical Equivalences Rosen 1.2

List of Logical Equivalences p  T  p; p  F  pIdentity Laws p  T  T; p  F  FDomination Laws p  p  p; p  p  p Idempotent Laws  (  p)  pDouble Negation Law p  q  q  p; p  q  q  pCommutative Laws (p  q)  r  p  (q  r); (p  q)  r  p  (q  r) Associative Laws

List of Equivalences p  (q  r)  (p  q)  (p  r)Distribution Laws p  (q  r)  (p  q)  (p  r)  (p  q)  (  p   q)De Morgan’s Laws  (p  q)  (  p   q) Miscellaneous p   p  TOr Tautology p   p  FAnd Contradiction (p  q)  (  p  q)Implication Equivalence p  q  (p  q)  (q  p)Biconditional Equivalence

The Proof Process Assumptions Logical Steps Conclusion (That which was to be proved) -Definitions -Already-proved equivalences -Statements (e.g., arithmetic or algebraic)

Prove: (p  q)  q  p  q (p  q)  q Left-Hand Statement  q  (p  q) Commutative  (q  p)  (q  q) Distributive  (q  p)  TOr Tautology  q  p Identity  p  q Commutative Begin with exactly the left-hand side statement End with exactly what is on the right Justify EVERY step with a logical equivalence

Prove: (p  q)  q  p  q (p  q)  q Left-Hand Statement  q  (p  q) Commutative  (q  p)  (q  q) Distributive Why did we need this step? Our logical equivalence specified that  is distributive on the right. This does not guarantee the equivalence works on the left! Ex.: Matrix multiplication is not always commutative (Note that whether or not  is distributive on the left is not the point here.)

Prove: p  q   q   p p  q   p  q Implication Equivalence  q   p Commutative   (  q)   p Double Negation   q   p Implication Equivalence Contrapositive

Prove: p  p  q is a tautology Must show that the statement is true for any value of p,q. p  p  q   p  (p  q) Implication Equivalence  (  p  p)  q Associative  (p   p)  q Commutative  T  q Or Tautology  q  T Commutative  T Domination This tautology is called the addition rule of inference.

Why do I have to justify everything? Note that your operation must have the same order of operands as the rule you quote unless you have already proven (and cite the proof) that order is not important. 3+4 = 4+3 3/4  4/3 A*B  B*A for everything (for example, matrix multiplication)

Prove: (p  q)  p is a tautology (p  q)  p   (p  q)  p Implication Equivalence  (  p  q)  pDeMorgan’s  (  q  p)  pCommutative   q  (  p  p)Associative   q  (p   p)Commutative   q  TOr Tautology  TDomination

Prove or Disprove p  q  p   q ??? To prove that something is not true it is enough to provide one counter-example. (Something that is true must be true in every case.) pqp  qp  q FTTFFTTF The statements are not logically equivalent

Prove:  p  q  p   q  p  q  (  p  q)  (q  p)Biconditional Equivalence  (  p  q)  (  q  p)Implication Equivalence (x2)  (p  q)  (  q  p)Double Negation  (q  p)  (  p  q)Commutative  (  q  p)  (  p  q)Double Negation  (  q  p)  (p  q)Implication Equivalence (x2)  p   q Biconditional Equivalence