Counterexample

A counterexample is generally used to check the validity of any statement. In order to disprove any arguments, theories, hypotheses, generalizations, statements, or theories. It can also be said as an example that meets the mathematical statement’s condition or hypothesis but does not lead to be true when concluding the given statement. 
 

Generally, counterexamples are used in the real life to contradict any particular statement if someone states them to be true. In order to identify a counterexample, let’s take an example of two students talking about the properties of “Square” during a geometry period at school.

Student A: “Any quadrilaterals with sides of equal length are square.”

Student B: “No, a rhombus has equal sides of the same length, but it is not a square.”

Here, Student A stated a hypothesis that any quadrilateral with equal sides is a square. But Student B put forward an example, that contradicted the statement that a rhombus also has equal sides, but it’s not square. Thus, Student B brought a counterexample to the statement done by Student A. 

Here are some tips to identify counterexamples:

• Any counterexample must be true for the statement or hypothesis, but it must be false for the conclusion.
  
   
• Counterexamples can demonstrate any statement as false, even if they appear to be true in some cases. For example, the statement “all prime numbers are odd numbers”. This might appear to be true at first glance, but the number 2 is an even prime number. So this can be stated as a hypothesis, with example of 2 as the counterexample. 
   

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

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

Let’s check this statement by calculating, 

     4 = 2 × 2 (Even)

     16 = 4 × 4 (Even)

     25 = 5 × 5 (Odd)

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

Conclusion: The given hypothesis is false. 

There is always room for doubt when it comes to counterexamples versus examples. Kids might think, why can't counterexamples be called as just examples? Here, let’s understand the basic difference between counterexamples and examples. 

Examples are provided when there needs to be support for any statement given. For example, “all prime numbers are odd.” Here, we can support this statement with an example that 3, 5, 7, 11, and 13 are all prime numbers which are also odd.

But on the other hand, counterexamples always contradict or refute the given statement. For example, “all the prime numbers are odd.” Here, we can disprove this statement by saying that prime number 2 is an even number. Thereby, giving a counterexample to the statement. Therefore, counterexamples are the opposite of examples. 
 

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 (True)

     20  5 = 4 (True)

     30  5 = 6 (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,

     4 is a multiple of 4 (True)

     6 is even but not a multiple of 4 

     8 is a multiple of 4  (True)

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

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

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 different-weight balls from the Leaning Tower of Pisa, and 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’s about the way they crack the exam.