Biconditional Math Example 4

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

Example 4

medium

Verify that

p \Leftrightarrow q \equiv (p \Rightarrow q) \land (q \Rightarrow p)

using a truth table.

Solution

1
Compute $p \Rightarrow q$ : $T,F,T,T$ . Compute $q \Rightarrow p$ : $T,T,F,T$ .
2
$(p \Rightarrow q) \land (q \Rightarrow p)$ : $T\land T, F\land T, T\land F, T\land T = T,F,F,T$ .
3
$p \Leftrightarrow q$ column is also $T,F,F,T$ . The columns match.

Answer

p \Leftrightarrow q \equiv (p \Rightarrow q) \land (q \Rightarrow p)

The biconditional is logically equivalent to requiring both directions of implication to hold simultaneously. This decomposition is how biconditionals are typically proved.

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 2 medium

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

Example 3 easy

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

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Related Concepts

Conditional Statement