Reversing a true statement does not keep it true. That is the whole lesson.
Judge the converse
"If a figure is a square, then it is a rectangle" is true. What about its converse?
The converse is "If a figure is a rectangle, then it is a square", and that is false — a 3 by 5 rectangle is a counterexample. A states the central misconception: a true statement's converse may be true or false, and must be checked separately. One counterexample is enough to settle it.
If hypothesis, then conclusion.
"If it rains, then the ground is wet."
a conditional
hypothesis: it rains
after "if"
conclusion: the ground is wet
after "then"
Many statements hide the if-then. "All squares are rectangles" means "If a figure is a square, then it is a rectangle" — rewriting it that way is often the first step.
| Name | Form | Example |
|---|---|---|
| Conditional | if p then q | If square, then rectangle ✓ |
| Converse | if q then p | If rectangle, then square ✗ |
| Inverse | if not p then not q | If not square, then not rectangle ✗ |
| Contrapositive | if not q then not p | If not rectangle, then not square ✓ |
A conditional and its contrapositive are always both true or both false. The converse and the inverse are a separate gamble — and notice they are contrapositives of each other, which is why they always match.
Proving a statement true takes an argument covering every case. Proving it false takes one example. That asymmetry is why counterexamples are so useful.
"If a triangle is equilateral, it is equiangular"
true
the converse is also true
equiangular → equilateral
so: equilateral if and only if equiangular
a biconditional
A definition is always a biconditional. "A square is a quadrilateral with four equal sides and four right angles" works in both directions — that is what makes it a definition rather than a mere fact.
Induction suggests what might be true and is how most discoveries start. Only deduction establishes it. Checking five cases proves nothing about the sixth.
The same swap-and-negate rules, where the wording makes them harder to see.
"All multiples of 6 are even." Write it as a conditional and test its converse.
If a number is a multiple of 6, then it is even.
true: 6 = 2 × 3
Converse: if a number is even, it is a multiple of 6.
swap
4 is even but not a multiple of 6
counterexample: false ✗
"All A are B" is "if A, then B" in disguise. Rewrite it before you swap anything.
The contrapositive of "If x > 5, then x² > 25"
not (x² > 25) means x² ≤ 25
the opposite of "greater than"
If x² ≤ 25, then x ≤ 5.
swap and negate: true ✓
Converse: if x² > 25, then x > 5
false: x = −6 gives 36 > 25
The negation of ">" is "≤", not "<" — the equal case belongs to the opposite side.
Is "a whole number is divisible by 10 if and only if it ends in 0" true?
divisible by 10 → ends in 0
true: 10, 20, 30 …
ends in 0 → divisible by 10
true: it is (tens) × 10
both directions hold
a true biconditional ✓
An "if and only if" needs both directions checked. One failure in either direction makes it false.
Expect a statement to turn into its converse, inverse or contrapositive, a question on which one is equivalent to the original, and a counterexample to pick (or the recognition that none exists). The fastest route: converse swaps, inverse negates, contrapositive does both, and only the contrapositive is guaranteed. The 10-second check for a counterexample: it must make the "if" part true and the "then" part false at the same time.
Write the contrapositive
What is the contrapositive of "If an angle is acute, then it measures less than 90°"?
The contrapositive both swaps and negates: "if not q, then not p". A is the converse (swap only) and B is the inverse (negate only). All three happen to be true here, because the original is a definition — but only C is the contrapositive, and only the contrapositive is guaranteed in general.
Hint: converse swaps, inverse negates, contrapositive does both. Work out which operations each option performed.
