Chapter 1
Number and Quantity
Course: Core Academic Skills For Educators Combined Test 5752
Subject: Core Academic Skills Educators Mathematics 5733
Chapter: Number And Quantity
1. Understanding Counterexamples
A counterexample is a specific case that proves a general statement to be false. In mathematics, a single counterexample is sufficient to disprove a statement, even if the statement is true in many other cases. This is because a universal claim — one that says something happens for all numbers or all situations — can only be valid if it holds without exception.
For example, the statement “All even numbers are divisible by 4” is not true because the number 6 is even but not divisible by 4. The number 6 meets the starting condition (being even) but does not meet the claim (divisibility by 4), making it a counterexample.
Counterexamples are important in mathematics because they encourage precise thinking, prevent overgeneralization, and help identify limitations of a rule or property.