Biconditional Math Example 1

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

easy

Evaluate the biconditional

p \Leftrightarrow q

for all truth value combinations and construct its truth table.

1
$p \Leftrightarrow q$ means ' $p$ if and only if $q$ ' — it is true when $p$ and $q$ have the same truth value.
2
Row $(T,T)$ : both true — same value — $T$ .
3
Row $(T,F)$ : different values — $F$ .
4
Row $(F,T)$ : different values — $F$ .
5
Row $(F,F)$ : both false — same value — $T$ .

Answer

\begin{array}{cc|c}p & q & p \Leftrightarrow q\\ \hline T&T&T\\T&F&F\\F&T&F\\F&F&T\end{array}

A biconditional is true precisely when both sides share the same truth value. It is equivalent to

(p \Rightarrow q) \land (q \Rightarrow p)

About Biconditional

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

Determine whether '[formula] is even [formula] [formula] is even' is true for all integers [formula]

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

Verify that [formula] using a truth table.