Examples of logically equivalent statements - College of Arts and Sciences

Propositional Logic Grinshpan

Examples of logically equivalent statements

Here are some pairs of logical equivalences. Each may be verified via a truth table.

pq

qp

commutativity of

pq

qp

commutativity of

?(?p)

p

double negation

(p q) r

p (q r)

associativity of

(p q) r

p (q r)

associativity of

p (q r)

(p q) (p r)

distributivity of over

p (q r)

(p q) (p r)

distributivity of over

?(p q)

?p ?q

de Morgans law

?(p q)

?p ?q

de Morgans law

p (p q)

p

absorption law

p (p q)

p

absorption law

pp

p

idempotency

pp

p

idempotency

pq

?p q

pq

?q ?q

pq

(p q) (q p)

Note that two statements P and Q are equivalent whenever P Q is a tautology. In the table below, T represents a statement that is always true and F represents a statement that is always false.

pT

p

pT

T

pF

F

pF

p

p ?p

F

p ?p

T

Precedence of logical operations

?

1

2

3

4

5

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