Write the contrapositive of: If n is even then n² is even

Question

Write the contrapositive of the statement: “If $n$ is even, then $n^2$ is even.” Also verify whether the original statement and its contrapositive are both true.

Solution — Step by Step

The statement “If $P$ , then $Q$ ” has the form $P \Rightarrow Q$ .

Here:

$P$ : ” $n$ is even”
$Q$ : ” $n^2$ is even”

The conditional statement is: $P \Rightarrow Q$ (If $P$ , then $Q$ )

The contrapositive of " $P \Rightarrow Q$ " is " $\lnot Q \Rightarrow \lnot P$ " (If NOT Q, then NOT P).

The negation of ” $n^2$ is even” is ” $n^2$ is not even” = ” $n^2$ is odd.”

The negation of ” $n$ is even” is ” $n$ is not even” = ” $n$ is odd.”

Therefore, the contrapositive is:

“If $n^2$ is odd, then $n$ is odd.”

If $n$ is even, then $n = 2k$ for some integer $k$ .

$n^2 = (2k)^2 = 4k^2 = 2(2k^2)$

Since $n^2 = 2 \times (2k^2)$ , it’s divisible by 2, so $n^2$ is even. ✓

The original statement is TRUE.

If $n^2$ is odd, we need to show $n$ is odd. We use proof by contradiction.

Assume $n^2$ is odd but $n$ is even. If $n$ is even, then (from Step 3) $n^2$ is even. But we assumed $n^2$ is odd — contradiction.

Therefore, if $n^2$ is odd, then $n$ must be odd. ✓

The contrapositive is TRUE.

A conditional statement and its contrapositive are logically equivalent — they always have the same truth value. If one is true, the other is necessarily true. This is why proving the contrapositive is a valid proof technique for the original statement.

Original: $P \Rightarrow Q$ (TRUE)
Contrapositive: $\lnot Q \Rightarrow \lnot P$ (TRUE) ✓

Why This Works

Contrapositive is one of the key logical equivalences. To prove $P \Rightarrow Q$ , sometimes it’s easier to prove $\lnot Q \Rightarrow \lnot P$ instead. They’re the same claim, just stated differently.

This is different from the converse ( $Q \Rightarrow P$ : “If $n^2$ is even, then $n$ is even”) and the inverse ( $\lnot P \Rightarrow \lnot Q$ : “If $n$ is odd, then $n^2$ is odd”). The converse and inverse are logically equivalent to each other, but NOT necessarily to the original statement.

Summary of all four related statements:

Statement	Form	Truth (for this example)
Original	$P \Rightarrow Q$	True
Converse	$Q \Rightarrow P$	True (but need to check separately)
Inverse	$\lnot P \Rightarrow \lnot Q$	True
Contrapositive	$\lnot Q \Rightarrow \lnot P$	True

(In this particular example, all four happen to be true, but this is not always the case.)

For CBSE Class 11 mathematical reasoning, expect questions asking you to write all four forms (original, converse, inverse, contrapositive) of a given statement. The fastest approach: identify P and Q clearly, negate each to get ¬P and ¬Q, then arrange into the four patterns.

Common Mistake

Students often confuse the contrapositive with the converse. The converse flips $P$ and $Q$ : “If $n^2$ is even, then $n$ is even.” The contrapositive negates BOTH and swaps them: “If $n^2$ is odd, then $n$ is odd.” The critical difference: the original statement and its contrapositive are always logically equivalent (same truth value). The original and its converse are NOT necessarily equivalent — they can have different truth values. Getting these mixed up is the most common error in this topic.

Question

Solution — Step by Step

Why This Works

Common Mistake

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