Two statements are said to be equivalent if they have the same truth value.

Example Following are two statements. p = It is false that he is a singer or he is a dancer. q = He is not a singer and he is not a dancer.

The first statement p consists of negation of two simple proposition a = He is a singer. b = He is a dancer. They are connected by an OR operator (connective) so we can write, p = ~(a ∨ b)

The second statement q consists of two simple proposition which are negation of a and b ~a = He is not a singer. ~b = He is not a dancer. They are connected by an AND operator (connective) so we can write, q = ~a ∧ ~b

We have, p = ~(a ∨ b) q = ~a ∧ ~b following are the truth tables for p and q

We can see that the truth values are same for both the statements. So, p and q are equivalent statements.

Now we will cover some equivalence laws.

Properties of 0

If x is a statement then, 0 + x = x 0 . x = 0 where + is the OR operator and . is the AND operator

Truth table

Properties of 1

If x is a statement then, 1 + x = 1 1 . x = x where + is the OR operator and . is the AND operator

Truth table

Involution

If p is a statement then, ~(~p) = p where ~ is the NOT operator

Truth table

Idempotence Law

If p is a statement then, p + p = p p . p = p where + is the OR operator and . is the AND operator

Truth table

Absorption Law

If p and q are two statements then, p + (p.q) = p p . (p + q) = p where + is the OR operator and . is the AND operator

Truth table

Complementarity Law

If p is a statement then, p + (~p) = 1 p . (~p) = 0 where + is the OR operator, . is the AND operator and ~ is the NOT operator

Truth table

Commutative Law

If p and q are two statements then, p + q = q + p p . q = q . p where + is the OR operator and . is the AND operator

Associative Law

If p, q and r are three statements then, (p + q) + r = p + (q + r) (p . q) . r = p . (q . r) where + is the OR operator and . is the AND operator

Distributive Law

If p, q and r are three statements then, p . (q + r) = (p . q) + (p . r) p + (q . r) = (p + q) . (p + r) p + (~p . q) = p + q where + is the OR operator, . is the AND operator and ~ is the NOT operator

De Morgan's Law

If p and q are two statements then, ~(p + q) = ~p . ~q ~(p . q) = ~p + ~q where + is the OR operator, . is the AND operator and ~ is the NOT operator

Truth table

Conditional Elimination

If p and q are two statements then, p ⇒ q = ~p + q where + is the OR operator and ~ is the NOT operator

Bi-conditional Elimination

If p and q are two statements then, p ⇔ q = (p . q) + (~p . ~q) where + is the OR operator, . is the AND operator and ~ is the NOT operator