Biconditional Statement

Biconditional Statement

How to solve biconditional statement problems: definition, its truth value, examples, and their solutions.

Definition, Truth Value

A biconditional statement is the conjunction of a conditional statement and its converse. A biconditional is true if both conditional and its converse are both true.

A biconditional statement is
the conjunction (∧) of
a conditional (pq) and its converse (qp).

pq is true
if both pq and qp are true.

Example 1

Find the truth value of the given statement: Angle A is a right angle if and only if the measure of angle A is 90.

p: ∠A is a right angle.
q: m∠A = 90.

pq:
If ∠A is a right angle, then m∠A = 90.

This is true.

qp:
If m∠A = 90, then ∠A is a right angle.

This is true.

pq and qp are both true.

So pq is true.

Example 2

Find the truth value of the given statement: x = 1 iff. x^2 = 1.

p: x = 1
q: x2 = 1

pq:
If x = 1, then x2 = 1.

This is true.
(12 = 1)

qp:
If x2 = 1, then x = 1.
This is false because there's a counterexample: -1.

If x = -1,
q is true: (-1)2 = 1
p is false: -1 ≠ 1
Then qp is false.
(True hypothesis and false conclusion → false conditional)

Conditional statement: truth value

pq is true.
qp is false.

So pq is false.