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 9
Q: the sum of the digits of the natural number is divisible by 9
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 9, then the sum of the digits of the natural number is divisible by 9.
- (Converse: "If Q then P" [P ← Q]) if the sum of the digits of the natural number is divisible by 9, then the natural number is divisible by 9.
- (Inverse: "If ~P then ~Q" [~P → ~Q]) if a natural number is NOT divisible by 9, then the sum of the digits of the natural number is NOT divisible by 9.
- (Contrapositive: "If ~Q then ~P" [~P ← ~Q]) if the sum of the digits of the natural number is NOT divisible by 9, then the natural number is NOT divisible by 9.