Counterexample

A counterexample is an example that meets all the conditions of a statement but does not give the expected result. This shows that the statement is not always true.

For example, let's consider a statement, “All even numbers are divisible by 4”.
 
This statement seems true at first, but the number 10 disproves it. 

As it meets the condition of being even, but it is not divisible by 4. 

Tips to identify a counterexample: 
 

• It must satisfy the hypothesis of the statement.
   
• It must contradict the conclusion.
   
• Even if a statement is often true, a single valid counterexample is enough to disprove it.

To find a counterexample, you look for a specific case that satisfies the condition of a statement but does not satisfy its conclusion. Here are the steps:
 

• Identify the hypothesis and conclusion of the statement.
   
• Check if a few examples satisfy the hypothesis.
   
• Check if any examples where the conclusion does not hold. If such an example exists, it is a counterexample, and the statement is not always true.

In order to find counterexamples, let’s take an example from mathematics.

Statement: If a number is a perfect square, then it is even. 

Let’s check this statement by calculating, 

  \(\implies 4 = 2 × 2 \) (Even)
 

  \(\implies 16 = 4 × 4\)  (Even)
 

  \(\implies 25 = 5 × 5 \) (Odd)

Here, 25 is the counterexample of this statement. 25 is a perfect square number, hence proving the hypothesis wrong.

Conclusion: The given hypothesis is false. 

In mathematics, both the counterexamples and regular examples serve different purposes. Here are the differences between a counterexample and a regular example. 
 

A conditional statement is an “if” statement. In counterexamples, we use them to test if a statement is true or false. If we find just one counterexample, then the statement is false. For example, 

Conditional Statements:

If a number is divisible by 10, then it is also divisible by 5.

Let’s check if it is true or false,

    \( 10 ÷ 5 = 2 {\text {(True)}} \\ \ \\ 20 ÷ 5 = 4 {\text {(True)}} \\ \ \\ 30 ÷ 5 = 6 {\text {(True)}}\)

Since every example is true, there is no counterexample, so the statement is true.

Conditional Statement:

If a number is even, then it is a multiple of 4.

Let’s check if it is true or false,

    \(\implies \) 4 is a multiple of 4 (True)
 

     \(\implies \) 6 is even but not a multiple of 4 

     \(\implies \) 8 is a multiple of 4 (True)

Here, 6 is a counterexample that proves the statement is false.
 

Counterexamples help students to understand how a single example can disprove a statement. Here are a few tips and tricks to master counterexamples. 

• To understand the statement, students can break it into conditions and conclusions, so they know precisely what must be checked. 
   
• Look for patterns, then try to find an exception that doesn’t follow the pattern.
   
• Start with simple values, then try exceptional cases such as 0, negative numbers, and fractions.
   
• Teachers can allow students to question statements and try different examples on their own.
   
• Parents can help their child explore counterexamples by encouraging trial-and-error learning. By asking questions like, “Can you think of an example that doesn’t fit?”

Counterexamples are not just used in math, they help us test ideas, concepts, science, and philosophies in real life too. Here are some real-life applications of counterexamples.

Philosophy: Back in history, Socrates provided a counterexample to Euthyphro’s definition of piety. This is called “Euthyphro” by Plato. Euthyphro suggested that whatever is pious is pleasing to the gods. But this was refuted by Socrates, pointing out that what is pleasing to one god will not be pleasing to another god. Thus disproving the definition. 

Science: Galileo dropped two balls of different weights from the Leaning Tower of Pisa. They hit the ground at the same time, thus disproving Aristotle’s old belief of gravity that “heavier objects fall faster than lighter ones”. 

Daily Life Myth: There is always a saying that “if you study a lot, you will always get good grades”. But this statement can be refuted, because a student’s grades are not dependent on how many hours they study. It depends on how effectively they approach the exam. 
 

Law: In legal reasoning, counterexamples are used to challenge claims or testimonies.


Geometry: Counterexamples are used to identify incorrect assumptions about shapes and their properties.