Contradiction

In his book A Mathematician’s Apology, G.H. Hardy identifies proof by contradiction as “one of a mathematician’s strongest weapons”. “It is a far finer gambit than any chess manoeuvre,” he continues, “since a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”

Proof by contradiction, often known as indirect evidence, is one of the most powerful methods of proof in mathematics. It may be used to prove any statement in various domains. The framework is: assume the proposition to be proven as false and try to demonstrate its untruth until the assumption leads to a contradiction.

How to Carry Out Proof by Contradiction?

Let’s break down the method of evidence by contradiction in steps to understand it better. The following is the procedure for using evidence by contradiction:

Step 1: Assume the statement’s opposite to be true (i.e., assume the statement is false).

Step 2: Begin an argument with the assumed assertion and work your way to the end.

Step 3: During this process, you should come into a paradox. This indicates that the alternative statement is wrong, and we can conclude that the original statement is correct.

This may appear complicated at first, so let’s look at some instances to help you grasp the concept.

Examples

Proving that the square root of 2 is irrational is a common example of proof by contradiction. Before we look at the evidence, there are a few definitions we need to know to understand it fully:

• Even number: a number m using the formula m = 2n, where n is an integer.

• Odd Number: a number r with the formula r = 2s + 1, where s is an integer. 14 is an even number, for example, since 14 = 2 * 7. 101 is also an odd number because it equals 2 * 50 + 1. It’s worth noting that if an integer isn’t even, it’s odd. It’s also true in the other direction. It is even if an integer is not odd.
• A rational number: a number with the form p/q, where p and q are integers. 3 and 0.9, for example, are rational numbers since 3 can be written as 3/1 and 0.9 as 9/10.

No Two Ways

The concepts of truth and falsity are opposites. If one exists, the other isn’t possible. This is a fundamental logical rule, and proof by contradiction is based on it. Since truth and falsity are mutually incompatible,

• It is impossible for a statement to be both true and untrue simultaneously.

• If a claim can be demonstrated to be true, it cannot be untrue.

• If a claim can be demonstrated to be untrue, it cannot be true.

• A statement is false if it cannot be proven true.

• If a statement can’t be demonstrated to be false, it’s true.

• The condition of truth and falsehood is exploited by evidence by contradiction, which is powerful and universal.

A Well-Known Contradiction

This is perhaps the most widespread example of contradiction:

√2 is irrational

Our proof will seek to disprove this claim. We shall make an effort to demonstrate this.

√2 is rational.

For the sake of argument, let us assume that the assertion ‘√2 is irrational’ is false.

You work till you discover the inconsistency.

A ratio or a fraction can be a rational number (numerator over denominator). Any fraction can be reduced to its irreducible form; for example, 2/6 can be reduced to 1/3 but not farther. At least one term of the fraction is odd in its simplified version.

A fraction of a ratio cannot be used to express an irrational number. Irrational numbers, such as Euler’s number e, have no fractional equivalent.

Since the square of every even integer is even, and every odd number is odd, we can see that a2 must be an even number. Because aa and bb can’t both be even, b has to be odd.)

Proof By Contradiction – Key Takeaways

The key steps involved in producing a proof by contradiction are:

Step 1: Take the statement and assume the opposite is correct (i.e. assume the statement is false).

Step 2: Begin an argument with the assumed assertion and work your way to the end.

Step 3: During this process, you should come into a paradox. This indicates that the alternative statement is wrong, and we can conclude that the original statement is correct.

There can only be two possible outcomes for the proposition we’re seeking to prove.

The idea behind proof by contradiction is that if the converse of a statement is always untrue, then the assertion must be true.

Conclusion

In logic and mathematics, proof by contradiction is a method of determining the truth of a statement by assuming it is false, then trying to show it is incorrect until the conclusion of that assumption is a contradiction. To be more specific, a mathematical theory is a set of sentences called theorems that are deduced using logical proofs. A contradiction is a phrase that includes its negation, therefore a theory that includes a contradiction is inconsistent.