Conditional Statement Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Conditional Statement.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A conditional P \to Q is a statement meaning "if P is true, then Q must be true," read as "if P then Q."

A promise or rule: if the condition holds, the consequence follows.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: P \to Q is false in exactly one situation: when P is true and Q is false. A broken promise requires the condition to actually be met.

Common stuck point: If P is false, P \to Q is automatically true (vacuously true).

Sense of Study hint: Ask yourself: 'Can I find a case where P is true but Q is false?' If yes, the implication fails. If no such case exists, it holds.

Worked Examples

Example 1

easy
Write the converse, inverse, and contrapositive of: 'If a number is divisible by 6, then it is divisible by 3.'

Solution

  1. 1
    Let p: 'a number is divisible by 6' and q: 'it is divisible by 3.' The original is p \Rightarrow q.
  2. 2
    Converse (q \Rightarrow p): 'If a number is divisible by 3, then it is divisible by 6.' (False; e.g., 9.)
  3. 3
    Inverse (\neg p \Rightarrow \neg q): 'If a number is not divisible by 6, then it is not divisible by 3.' (False; e.g., 9.)
  4. 4
    Contrapositive (\neg q \Rightarrow \neg p): 'If a number is not divisible by 3, then it is not divisible by 6.' (True.)

Answer

\text{Contrapositive: If not divisible by 3, then not divisible by 6 (true).}
A conditional p \Rightarrow q is logically equivalent to its contrapositive \neg q \Rightarrow \neg p, but not necessarily to its converse or inverse.

Example 2

medium
Determine the truth value of: 'If 2 > 5, then 10 > 3.'

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Identify the hypothesis and conclusion in: 'If it rains, then the ground is wet.'

Example 2

medium
Identify the hypothesis and the conclusion in the statement: 'If a number is divisible by 4, then it is even.'
← Back to Conditional Statement Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

logical statement