A natural number is divisible by 3 if and only if the sum of the digits of the natural number is divisible by 3.

The biconditional sentence above compactly states the following four conditional sentences:

• if a natural number is divisible by 3, then the sum of the digits of the natural number is divisible by 3.
• if the sum of the digits of the natural number is divisible by 3, then the natural number is divisible by 3.
• if a natural number is NOT divisible by 3, then the sum of the digits of the natural number is NOT divisible by 3.
• if the sum of the digits of the natural number is NOT divisible by 3, then the natural number is NOT divisible by 3.

If "P" represents one statement and "Q" represents another statement, then "If P then Q" is called a conditional sentence symbolized as "P → Q"

The following four conditional sentences can be considered:

The biconditional sentence P if and only if Q states that the conditional sentence, its converse, inverse and contrapositive are all true.

The biconditional sentence of this example can be broken up into the two parts below.

P: natural number is divisibly by 3

Q: the sum of the digits of the natural number is divisible by 3

P if and only if Q [i.e. P ↔ Q] results in the following:

• (Conditional sentence: "If P then Q" [P → Q]) if a natural number is divisible by 3, then the sum of the digits of the natural number is divisible by 3.
• (Converse: "If Q then P" [P ← Q]) if the sum of the digits of the natural number is divisible by 3, then the natural number is divisible by 3.
• (Inverse: "If ~P then ~Q" [~P → ~Q]) if a natural number is NOT divisible by 3, then the sum of the digits of the natural number is NOT divisible by 3.
• (Contrapositive: "If ~Q then ~P" [~P ← ~Q]) if the sum of the digits of the natural number is NOT divisible by 3, then the natural number is NOT divisible by 3.