Definition: A conditional statement has the form if p, then q; it is false only when p is true and q is false, and it has a converse, inverse, and contrapositive.
What are its parts?
The hypothesis p follows if, and the conclusion q follows then. Example: If a polygon is a square, then it has four sides. Here p is a polygon is a square, and q is it has four sides.
What are the related statements?
- Converse: if q, then p.
- Inverse: if not p, then not q.
- Contrapositive: if not q, then not p.
A conditional and its contrapositive always have the same truth value. The converse and inverse can differ from the original.
Example. If a number is divisible by 6, then it is even. True. Converse: if a number is even, then it is divisible by 6. False; 10 is a counterexample. Contrapositive: if a number is odd, then it is not divisible by 6. True.
What mistakes occur?
Assuming the converse is true whenever the original is. Another is thinking a counterexample must satisfy the conclusion; it must satisfy the hypothesis while failing the conclusion.