# 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 (*p* → *q*) and its converse (*q* → *p*).*p* ↔ *q* is true

if both *p* → *q* and *q* → *p* are true.

## Example 1

*p*: ∠*A* is a right angle.*q*: m∠*A* = 90.*p* → *q*:

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

This is true.

*q* → *p*:

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

This is true.

*p* → *q* and *q* → *p* are both true.

So *p* ↔ *q* is true.

## Example 2

*p*: *x* = 1*q*: *x*^{2} = 1*p* → *q*:

If *x* = 1, then *x*^{2} = 1.

This is true.

(1^{2} = 1)

*q* → *p*:

If *x*^{2} = 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 *q* → *p* is false.

(True hypothesis and false conclusion → false conditional)

Conditional statement: truth value

*p* → *q* is true.*q* → *p* is false.

So *p* ↔ *q* is false.