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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