Biconditional Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium

Determine whether '

n

is even

\Leftrightarrow

n^2

is even' is true for all integers

n

Solution

1
Forward direction ( $\Rightarrow$ ): If $n$ is even, $n = 2k$ , so $n^2 = 4k^2 = 2(2k^2)$ , which is even. True.
2
Backward direction ( $\Leftarrow$ ): If $n^2$ is even, then $n$ must be even (by contrapositive: if $n$ odd, $n^2$ odd — proved earlier). True.
3
Both directions hold, so the biconditional is true for all integers $n$ .

Answer

n \text{ is even} \Leftrightarrow n^2 \text{ is even (true for all integers)}

To prove a biconditional, prove both the forward and backward conditionals. This example also illustrates how the contrapositive aids the backward direction.

About Biconditional

A biconditional $P \leftrightarrow Q$ is true when $P$ and $Q$ have the same truth value — both true or both false.

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More Biconditional Examples

Example 1 easy

Evaluate the biconditional [formula] for all truth value combinations and construct its truth table.

Example 3 easy

State whether each biconditional is true or false: (a) '[formula]', (b) '[formula]', (c) '[formula]'

Example 4 medium

Verify that [formula] using a truth table.

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Related Concepts

Conditional Statement