Proof by Contradiction

Generally, we use direct arguments to prove the truth of the statement. Sometimes, we use indirect arguments to prove the statement using a powerful tool in Mathematics called the “Proof by Contradiction”. In this article, we will discuss how to prove the statement using the proof by contradiction method with the help of an example.

What is Meant by Proof by Contradiction?

In Mathematics, a contradiction occurs when we get a statement p, such that p is true and its negation ~p is also true. Now, let us understand the concept of contradiction with the help of an example.

Consider two statements p and q.

Statement p: x = a/b, where a and b are co-prime numbers.

Statement q: 2 divides both “a” and “b”.

Here, in this case, we have to assume that the statement “p” is true” and also manage to show that the statement “q” is true. Hence, we have arrived at a contradiction, because the statement “q’ implies the negation of statement “p” is true.

Proof by Contradiction Example

Question:

Show that the product of a non-zero rational number and an irrational number is an irrational number using the proof by contradiction method.

Solution:

Now, we will use the method called “ proof by contradiction” to show that the product of a non-zero rational number and an irrational number is an irrational number.

Statement

Comments

Let “r” be a non-zero rational number and x be an irrational number.

Assume that r= m/n, where m and n are integers, where m≠ 0, and n≠ 0.

So, we need to prove rx is irrational

Assume that rx is rational.

Take the negation of a statement that we need to prove.

Hence, rx = p/q, where p and q are integers, and q≠0.

Following the previous statement and from the definition of the rational number.

Now, by rearranging the equation, rx = p/q, and q≠0, and by using the fact r=m/n, we get x = p/rq = np/mq.

As “np” and “mq” are integers and mq≠0, x is a rational number

Using the properties of integers and by using the definition of rational numbers.

Hence, the result is a contradiction, because we have proved that x is rational, but by our hypothesis, we have x is irrational.

This is what we called the proof by contradiction.

The contradiction has arisen due to the faulty assumption that rx is rational. Hence, rx is irrational.

From logical deduction.

Hence, the given statement is proved using the proof by contradiction method.

Practice Problems

  1. Prove that if for an integer a, a2 is divisible by 3, then a is divisible by 3 using the proof by contradiction.
  2. Assume that r is a rational number and x is an irrational number. Prove that r+x is an irrational number using the proof by contradiction.
  3. Let a+b = c+d, and a<c. Show that b>d using the proof by contradiction method.

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