Conditional statements are a basic structure in mathematics: they connect a hypothesis to a conclusion using the form if p, then q. They matter because many definitions, theorems, and proofs are written as conditionals. Learning how to read and transform these statements helps students decide what a theorem actually says and what it does not say.
This skill also supports clear reasoning in geometry, algebra, computer science, and everyday arguments.
A conditional statement p -> q has three common related forms: the converse q -> p, the inverse not p -> not q, and the contrapositive not q -> not p. The original conditional and its contrapositive are logically equivalent, meaning they always have the same truth value. The converse and inverse are also equivalent to each other, but not necessarily to the original statement.
A single counterexample can disprove a conditional by showing a case where p is true but q is false.
Key Facts
- Conditional: p -> q means if p, then q.
- Converse: q -> p switches the hypothesis and conclusion.
- Inverse: not p -> not q negates both parts of the original conditional.
- Contrapositive: not q -> not p switches and negates both parts.
- Logical equivalence: p -> q is equivalent to not q -> not p.
- A conditional p -> q is false only when p is true and q is false.
Vocabulary
- Conditional statement
- A statement in the form if p, then q, where p is the hypothesis and q is the conclusion.
- Hypothesis
- The part of a conditional statement that follows if and gives the condition being assumed.
- Conclusion
- The part of a conditional statement that follows then and gives the result claimed.
- Contrapositive
- The statement formed by switching and negating the hypothesis and conclusion of a conditional.
- Counterexample
- A specific example that proves a statement false by satisfying the hypothesis but not the conclusion.
Common Mistakes to Avoid
- Treating the converse as automatically true is wrong because q -> p does not always have the same truth value as p -> q.
- Forgetting to negate both parts when forming the inverse is wrong because the inverse of p -> q must be not p -> not q.
- Using an example that makes both p and q true as a proof is wrong because examples support a pattern but do not prove a universal conditional.
- Giving a counterexample where the hypothesis is false is wrong because a counterexample must make p true and q false.
Practice Questions
- 1 Let p be true and q be false. Find the truth values of p -> q, q -> p, not p -> not q, and not q -> not p.
- 2 A truth table has 4 possible rows for p and q. In how many rows is p -> q true, and in how many rows is it false?
- 3 For the statement if a number is divisible by 4, then it is even, write the converse, inverse, and contrapositive. Identify which related statement is guaranteed to be logically equivalent to the original.