Negate the statement: All prime numbers are odd

Question

Write the negation of the statement: “All prime numbers are odd.” Is the negation true or false? Justify your answer.

Solution — Step by Step

The statement “All prime numbers are odd” is a universal statement of the form:

\forall x: P(x) \Rightarrow Q(x)

In plain English: “For every $x$ , if $x$ is a prime number, then $x$ is odd.”

Here, $P(x)$ = ” $x$ is prime” and $Q(x)$ = ” $x$ is odd.”

The negation of “For all $x$ , $P(x) \Rightarrow Q(x)$ ” is:

\exists x: P(x) \land \neg Q(x)

In plain English: “There exists some $x$ that is prime but NOT odd.”

So the negation is: “There exists a prime number that is not odd” (i.e., that is even).

Negation: “There exists at least one prime number that is not odd.”

Or equivalently: “Some prime number is even.”

Or more directly: “Not all prime numbers are odd.”

All three phrasings are logically equivalent.

To check the truth value, we need a counterexample to the original statement — a prime number that is not odd (i.e., an even prime).

The number 2 is:

Even (not odd)
Prime (divisible only by 1 and itself)

2 is the only even prime. Every other even number has 2 as a factor, so it’s composite.

Since 2 is a prime that is not odd, the negation is TRUE.

Therefore, the original statement “All prime numbers are odd” is FALSE.

Why This Works

In mathematical logic, negating a universal statement (“all $P$ are $Q$ ”) always gives an existential statement (“there exists a $P$ that is not $Q$ ”). You only need one counterexample to prove a universal statement false.

This is the principle of proof by counterexample: a single exception disproves a universal claim. We don’t need to show that most primes are even — just one (which is 2) is enough.

The negation of “All A are B” is “Some A are not B” (not “All A are not B”). Students confuse negation with the contrary. “No prime is odd” would be the contrary, not the negation. The negation is weaker — it just says at least one exception exists.

Alternative Method

Using De Morgan’s laws in set language: the complement of the set “all primes are odd” corresponds to the set of primes that fall outside the odd numbers. That set is {2} — non-empty, confirming the negation is true.

Common Mistake

A very common error: writing the negation as “All prime numbers are even” or “No prime number is odd.” These are stronger claims and are not the correct logical negation. The negation only asserts that the universal claim fails — i.e., that at least one prime is not odd. It makes no claim about all primes or most primes.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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