Conditional statements
Most geometric facts are if-then statements. Knowing how to rearrange them, and which rearrangements stay true, prevents some of the most common errors in reasoning.
Conditionals
A conditional statement has the form 'if p, then q', written p implies q. The part after 'if' is the hypothesis; the part after 'then' is the conclusion. 'If an angle measures 90 degrees, then it is a right angle' has hypothesis 'an angle measures 90 degrees' and conclusion 'it is a right angle'.
Truth of a conditional
A conditional is false only when the hypothesis is true and the conclusion false. 'If it is raining, then the ground is wet' is broken only by rain on dry ground. Statements in everyday language can often be rewritten in if-then form: 'All squares are rectangles' becomes 'If a figure is a square, then it is a rectangle'.
Converse
The converse swaps hypothesis and conclusion: 'if q, then p'. A true conditional can have a false converse. 'If a figure is a square, then it has four sides' is true, but its converse, 'if a figure has four sides, then it is a square', is false: a trapezoid is a counterexample.
Inverse and contrapositive
The inverse negates both parts: 'if not p, then not q'. The contrapositive swaps and negates: 'if not q, then not p'. A conditional and its contrapositive are always both true or both false; they are logically equivalent. The converse and inverse are equivalent to each other.
Biconditionals
When a conditional and its converse are both true, they combine into a biconditional using 'if and only if'. 'An angle is right if and only if it measures 90 degrees' is a biconditional. Good definitions can always be written as biconditionals.
Using the contrapositive
Because a statement and its contrapositive are equivalent, proving the contrapositive proves the original. To show 'if n squared is even, then n is even', it is easier to prove 'if n is odd, then n squared is odd': (2k + 1) squared = 4k squared + 4k + 1, which is odd.
Common fallacies
Affirming the conclusion is the error of reasoning from q back to p: 'If a figure is a square it has four right angles; this figure has four right angles; so it is a square' is invalid. Denying the hypothesis is the error 'if not p, then not q'. Advertising and arguments often rely on these fallacies.
Worked example
Statement: if two angles are vertical angles, then they are congruent (true). Converse: if two angles are congruent, then they are vertical angles (false; two separate 40-degree angles are a counterexample). Inverse: if two angles are not vertical, then they are not congruent (false). Contrapositive: if two angles are not congruent, then they are not vertical (true).
Key takeaways
A conditional 'if p, then q' has a hypothesis p and conclusion q and is false only when p is true and q false. The converse swaps the parts, the inverse negates both, and the contrapositive swaps and negates. A statement and its contrapositive are logically equivalent; the converse and inverse are equivalent to each other but not to the original. True statements with true converses form biconditionals ('if and only if'). Avoid affirming the conclusion and denying the hypothesis.
Flashcard
If p, then q
Check your understanding
Which statement is logically equivalent to 'If a figure is a square, then it has four sides'?