Proofs Using Logical Equivalences

Rosen 1.2

List of Logical EquivalencespT p; pF p Identity Laws

pT T; pF F Domination Laws

pp p; pp p Idempotent Laws

(p) p Double Negation Law

pq qp; pq qp Commutative Laws

(pq) r p (qr); (pq) r p (qr) Associative Laws

List of Equivalencesp(qr) (pq)(pr) Distribution Lawsp(qr) (pq)(pr)

(pq)(p q) De Morgan’s Laws(pq)(p q)

Miscellaneousp p T Or Tautologyp p F And Contradiction(pq) (p q) Implication Equivalence

pq(pq) (qp) 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: (pq) q pq (pq) q Left-Hand Statement

q (pq) Commutative

(qp) (q q) Distributive

(qp) T Or Tautology

qp Identity

pq Commutative

Begin with exactly the left-hand side statementEnd with exactly what is on the right

Justify EVERY step with a logical equivalence

Prove: (pq) q pq (pq) q Left-Hand Statement

q (pq) Commutative(qp) (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 tautologyMust 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 CommutativeT DominationThis 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: (pq) p is a tautology

(pq) p

(pq) p Implication Equivalence

(pq) p DeMorgan’s

(qp) p Commutative

q (p p) Associative

q (p p) Commutative

q T Or Tautology

T Domination

Prove or Disprovep 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.)

p q pq pqF T T F

The statements are not logically equivalent

Prove:p q p q

p q

(pq) (qp) Biconditional Equivalence

(pq) (qp)Implication Equivalence (x2)

(pq) (qp) Double Negation

(qp) (pq) Commutative

(qp) (pq)Double Negation

(qp) (pq) Implication Equivalence (x2)

p q Biconditional Equivalence