# Disproof by Counterexample

Disproof by counterexampleÂ is the technique in mathematics where a statement is shown to be wrong by finding a single example for when it is not satisfied. Not surprisingly, disproof is the opposite of proof so instead of showing that something is true, we must show that it is false. Any statement that makes inferences about a set of numbers can be disproved by finding a single example for which it does not work.

## Examples

Prove or disprove the statement that all prime numbers are odd.

Solution:

At first thought, it might seem that all prime numbers are odd. This is because it seems that all even numbers are not prime as 2 is a factor. However, by definition, 2 is a prime number but it is not odd and so we have found an example of when the statement is not true. This disproves the statement by counterexample.

Disprove by counterexample that for any , if , then

Solution:

Note that is the set of all positive or negative integers. Finding an and such that but , then the statement is disproved. Choosing any integer for and then choosing will accomplish this. For example, let and . In this case and and so we have found an example where but and thus disproving the statement.

Find any values for and for which the following statement isn’t true:

Solution:

The tendency in this situation is to try various positive values of and until the statement isn’t true. However, the statement IS true for all positive values and so negative values of and must be tried. Bear in mind that you can only square root a positive number (without complex numbers) and so both and must both be negative. Try, for example, and . The left hand side is the square root of which is whereas the right hand side is . Since neither 4 or -4 are less than or equal to , we have found a counterexample.

## Videos

This video shows the solutions that a student should give when tackling a basic disproof by counterexample question.

This video explores a proof exam question where the first part is a standard Proof by Deduction question and the second part requires the student to decide which proof to use – it turns out the statement isn’t true and a simple counterexample will suffice to disprove the statement.
The text passage contains multiple examples of mathematical statements being disproved by counterexample. Each example presents a statement and then provides a specific counterexample to show that the statement is not always true.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Disproof by Counterexample

Disproof by counterexampleÂ is the technique in mathematics where a statement is shown to be wrong by finding a single example for when it is not satisfied. Not surprisingly, disproof is the opposite of proof so instead of showing that something is true, we must show that it is false. Any statement that makes inferences about a set of numbers can be disproved by finding a single example for which it does not work.

## Examples

Prove or disprove the statement that all prime numbers are odd.

Solution:

At first thought, it might seem that all prime numbers are odd. This is because it seems that all even numbers are not prime as 2 is a factor. However, by definition, 2 is a prime number but it is not odd and so we have found an example of when the statement is not true. This disproves the statement by counterexample.

Disprove by counterexample that for any , if , then

Solution:

Note that is the set of all positive or negative integers. Finding an and such that but , then the statement is disproved. Choosing any integer for and then choosing will accomplish this. For example, let and . In this case and and so we have found an example where but and thus disproving the statement.

Find any values for and for which the following statement isn’t true:

Solution:

The tendency in this situation is to try various positive values of and until the statement isn’t true. However, the statement IS true for all positive values and so negative values of and must be tried. Bear in mind that you can only square root a positive number (without complex numbers) and so both and must both be negative. Try, for example, and . The left hand side is the square root of which is whereas the right hand side is . Since neither 4 or -4 are less than or equal to , we have found a counterexample.

## Videos

This video shows the solutions that a student should give when tackling a basic disproof by counterexample question.

This video explores a proof exam question where the first part is a standard Proof by Deduction question and the second part requires the student to decide which proof to use – it turns out the statement isn’t true and a simple counterexample will suffice to disprove the statement.
user
The text passage contains multiple examples of mathematical statements being disproved by counterexample. Each example presents a statement and then provides a specific counterexample to show that the statement is not always true.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS