COUNTER EXAMPLE

Counter Example

Definition Of Counter Example

Counterexample is an example that is used to disprove a statement.

Example of Counter Example

The opposite of a number is always positive.
Counterexample for the above statement: The opposite of 2 is - 2, a negative number.

Video Examples: Geometry - Inductive Reasoning & Counterexamples

Solved Example on Counter Example

Ques: Which of the following is a counterexample of the statement, "The perimeter of a rectangle can never be an odd number"?

Choices:

A. Length of a rectangle is 3.25 in. and width is 5.25 in.
B. Length of a rectangle is 10 in. and width is 7 in.
C. Length of a rectangle is 6.5 in. and width is 4.5 in.
D. Length of a rectangle is 9 in. and width is 5 in.
Correct Answer: A

Solution:

Step 1: The counterexample that disproves the given statement is:
'Length of a rectangle is 3.25 in. and width is 5.25 in.'

Quick Summary

A counterexample disproves a statement.
It only takes one counterexample to disprove a universal statement.

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🍎 Teacher Insights

Emphasize the importance of rigorous proof over intuitive examples. Use real-world examples to illustrate how counterexamples can reveal flaws in reasoning.

🎓 Prerequisites

Basic algebra
Understanding of mathematical statements

Check Your Knowledge

Q1: Which of the following is a counterexample to the statement 'All prime numbers are odd'?

2 3 5 7

Q2: Which of the following is a counterexample of the statement, 'The perimeter of a rectangle can never be an odd number'?

Length of a rectangle is 3.25 in. and width is 5.25 in. Length of a rectangle is 10 in. and width is 7 in. Length of a rectangle is 6.5 in. and width is 4.5 in. Length of a rectangle is 9 in. and width is 5 in.

Frequently Asked Questions

Q: How many counterexamples are needed to disprove a statement?
A: Only one counterexample is needed.

Q: Can a statement be proven true by finding examples?
A: No, examples can suggest a statement is true, but they do not prove it. Only a formal proof can establish truth.