Contrapositive

Q: What is the difference between the contrapositive and the converse?

The converse of $p \Rightarrow q$ is $q \Rightarrow p$ — you swap the hypothesis and conclusion but do not negate. The contrapositive is $\neg q \Rightarrow \neg p$ — you both swap and negate. The converse is not necessarily equivalent to the original, but the contrapositive always is.

Contrapositive

Switching the hypothesis and conclusion of a conditional statement and negating both. For example, the contrapositive of "If it is raining then the grass is wet" is "If the grass is not wet then it is not raining."

Note: As in the example, the contrapositive of any true proposition is also true.

Key Formula

\text{Original: } p \Rightarrow q \qquad \text{Contrapositive: } \neg q \Rightarrow \neg p

Where:

$p$ = The hypothesis (the "if" part of the original statement)
$q$ = The conclusion (the "then" part of the original statement)
$\neg$ = Negation symbol, meaning "not"
$\Rightarrow$ = "Implies" or "if … then"

Example

Problem: Write the contrapositive of the statement: "If a number is divisible by 6, then it is divisible by 2." Then determine whether the contrapositive is true.

Step 1: Identify the hypothesis (p) and conclusion (q).

p: \text{a number is divisible by 6} \qquad q: \text{a number is divisible by 2}

Step 2: Negate both parts.

\neg q: \text{a number is NOT divisible by 2} \qquad \neg p: \text{a number is NOT divisible by 6}

Step 3: Swap the order so the negated conclusion comes first, forming the new "if" clause.

\neg q \Rightarrow \neg p

Step 4: Write the contrapositive in words: "If a number is not divisible by 2, then it is not divisible by 6."

Step 5: Check the truth value. If a number is not divisible by 2, it is odd. An odd number can never be divisible by 6 (since 6 = 2 × 3 requires divisibility by 2). So the contrapositive is true — which is expected, because the original statement is true.

Answer: The contrapositive is: "If a number is not divisible by 2, then it is not divisible by 6." It is true.

Another Example

Problem: Write the contrapositive of: "If a triangle is equilateral, then all its angles measure 60°."

Step 1: Identify the parts. Hypothesis: a triangle is equilateral. Conclusion: all its angles measure 60°.

Step 2: Negate both and swap their positions.

Step 3: The contrapositive is: "If a triangle does not have all angles measuring 60°, then it is not equilateral."

\neg q \Rightarrow \neg p

Answer: "If a triangle does not have all angles measuring 60°, then it is not equilateral." This is logically equivalent to the original true statement.

Frequently Asked Questions

Is the contrapositive always true if the original statement is true?

Yes. A conditional statement and its contrapositive are logically equivalent, meaning they always share the same truth value. If the original is true, the contrapositive is true; if the original is false, the contrapositive is false.

What is the difference between the contrapositive and the converse?

The converse of

p \Rightarrow q

q \Rightarrow p

— you swap the hypothesis and conclusion but do not negate. The contrapositive is

\neg q \Rightarrow \neg p

— you both swap and negate. The converse is not necessarily equivalent to the original, but the contrapositive always is.

Contrapositive vs. Converse

Given the conditional

p \Rightarrow q

: the contrapositive is

\neg q \Rightarrow \neg p

(swap and negate both parts), while the converse is

q \Rightarrow p

(swap without negating). The contrapositive is always logically equivalent to the original statement. The converse is not — it may be true or false independently. For example, "If it rains, the ground is wet" is true. Its contrapositive "If the ground is not wet, it is not raining" is also true. But its converse "If the ground is wet, it is raining" could be false (the sprinkler might be on).

Why It Matters

The contrapositive is a powerful tool in mathematical proofs. When proving a statement

p \Rightarrow q

directly is difficult, you can instead prove its contrapositive

\neg q \Rightarrow \neg p

, which is logically the same thing. This technique — called proof by contrapositive — appears throughout algebra, geometry, and number theory and is one of the first formal proof strategies students learn.

Common Mistakes

Mistake: Confusing the contrapositive with the converse. Students often just swap p and q without negating, producing the converse instead.

Correction: The contrapositive requires two operations: swap the hypothesis and conclusion AND negate both. If you only swap, you have the converse, which is not logically equivalent to the original.

Mistake: Confusing the contrapositive with the inverse. Students sometimes negate both parts but forget to swap them, writing

\neg p \Rightarrow \neg q

Correction: That is the inverse of the conditional, not the contrapositive. The contrapositive must both swap and negate:

\neg q \Rightarrow \neg p

. The inverse is equivalent to the converse, not to the original statement.

Related Terms

Conditional — The if-then statement the contrapositive is formed from
Converse — Swaps hypothesis and conclusion without negating
Inverse of a Conditional — Negates both parts without swapping
Hypothesis — The 'if' part of a conditional statement
Conclusion — The 'then' part of a conditional statement
If and Only If — Biconditional combining a statement and its converse
Negation — The 'not' operation used to form the contrapositive