What exactly is assumed in the proof by contradiction for this theorem?

Conditions

• The theorem to be proved is: the product of two even numbers is also even.
• The method is proof by contradiction.
• The parity dichotomy (not even implies odd) is used as an assumed fact.

Reasoning, step by step

1. State the goal: prove that the product of two even numbers is even.
2. Introduce the method: proof by contradiction.
3. Write the negation of the desired conclusion: "Assume that the product of two even numbers is odd."
4. Justify the negation by noting that if it is not even, it must be odd.

Example

The presenter writes "Assume that the product of two even numbers is odd." The speaker says, "we'll again we'll use a proof by contradiction. So PBC. So we're going to assume the opposite." and "So if it's not even, it must be odd."

Common misconceptions

• Assuming the product is neither even nor odd.
• Thinking the assumption is that the product is even, which is the conclusion to be proved.

Watch the explanation

Explore next

• How are the arbitrary even numbers represented algebraically in the proof?
• Why does the presenter switch from examples like $6\times 8$ and $2\times 4$ to an algebraic proof?
• How is the product of two even numbers rewritten to show it is twice another quantity?