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# Counterexamples | Lesson

## What is a counterexample?

A mathematical statement is a sentence that is either true or false.
A mathematical statement has two parts: a condition and a conclusion.
Showing that a mathematical statement is true requires a formal proof.
However, showing that a mathematical statement is false only requires finding one example where the statement isn't true. Such an example is called a counterexample because it's an example that counters, or goes against, the statement's conclusion.

### What skills are tested?

• Identifying a counterexample to a mathematical statement

## How can we identify counterexamples?

When identifying a counterexample, follow these steps:
1. Identify the condition and conclusion of the statement.
2. Eliminate choices that don't satisfy the statement's condition.
3. For the remaining choices, counterexamples are those where the statement's conclusion isn't true.

TRY: IDENTIFYING A COUNTEREXAMPLE
The square of an integer is always an even number.
Which of the following numbers provides a counterexample showing that the statement above is false?

TRY: IDENTIFYING A COUNTEREXAMPLE
Which of the following expressions can be used to show that the sum of two numbers is not always greater than both numbers?

TRY: IDENTIFYING COUNTEREXAMPLES
Bart claims that all numbers that are multiples of 5 are also multiples of 10. Which of the following numbers can be used to show that Bart's statement is not true?

TRY: IDENTIFYING A COUNTEREXAMPLE
A student claims that when any two even numbers are multiplied, all of the digits in the product are even. Which of the following shows that the student is wrong?

## Things to remember

A counterexample to a mathematical statement is an example that satisfies the statement's condition(s) but does not lead to the statement's conclusion.
Identifying counterexamples is a way to show that a mathematical statement is false.
When identifying a counterexample,
1. Identify the condition and conclusion of the statement.
2. Eliminate choices that don't satisfy the statement's condition.
3. For the remaining choices, counterexamples are those where the statement's conclusion isn't true.

## Want to join the conversation?

• Does a counter example have to an equation or can we use words and sentences?
• You will need to use words to describe why the counter example you've chosen satisfies the "condition" (aka "hypothesis"), but does not satisfy the "conclusion"
Example: Tell whether the statement is True or False, then if it is false, find a counter example:
If a number is a rational number, then the number is positive.
Solution: This statement is false, -5 is a rational number but not positive.