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One of several different ways to prove a statement in mathematics is proof by contradiction. Learn the definition of this method and observe how it is applied to proving a statement’s truth value through examples and exploration.

Proof By Contradiction

In the book A Mathematician’s Apology by G.H. Hardy (pictured below), he describes proof by contradiction as ‘one of a mathematician’s finest weapons.’ He went on to say, ‘It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.’Sometimes, when it is difficult or downright impossible to prove the truth of a statement directly, we may turn to proof by contradiction.

We know that a statement cannot both be true and false – it has to be one or the other. Proof by contradiction uses this fact to prove something is true by showing that it cannot be false. When proving something is true using proof by contradiction, you assume the statement to be false, and as you proceed with the proof, you run into a contradiction, making the assumption of your original statement being false impossible, thus it must be true.

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When deciding if proof by contradiction is the best way to prove a given statement, it is a good idea to ask yourself, what would happen if the statement weren’t true? If the result of the statement not being true leads to something that makes no sense, such as 1 = 5 or p is even and p is odd, then proof by contradiction is a good way to proceed.

Steps

To clarify the process of proof by contradiction further, let’s break it down into steps. When using proof by contradiction, we follow these steps.

  1. Assume your statement to be false.
  2. Proceed as you would with a direct proof.
  3. Come across a contradiction.
  4. State that because of the contradiction, it can’t be the case that the statement is false, so it must be true.

Examples

A very common example of proof by contradiction is proving that the square root of 2 is irrational. Before looking at this proof, there are a few definitions we will need to know in order to understand the proof:

  • Even number: a number m that can be written as m = 2n where n is an integer
  • Odd number: a number r that can be written as r = 2s + 1, where s is an integer. For example, 14 is an even number because 14 = 2 * 7. Similarly, 101 is an odd number because 101 = 2 * 50 + 1.

    Notice that if an integer is not even, then it is odd. The reverse is also true. If an integer is not odd, then it is even.

  • Rational number: a number that can be written as p/q where p and q are integers.

    For example, 3 and 0.9 are rational numbers because we can write 3 as 3/1 and we can write 0.9 as 9/10.

Now, let’s take a look at this proof that the square root of 2 is irrational.Statement: The square root of 2 is irrational.

Proof by contradiction:Assume not. That is, assume that the square root of 2 is rational. Then the square root of 2 can be written as p/q where p and q are integers, and p/q is reduced as much as possible (p and q don’t share any common factors). Observe:

proof that square root of 2 is irrational - 2

Since p^2 = 2q^2, and p^2 = 4k^2, we see that 2q^2 = 4k^2. If we divide both sides of this by 2, we have that q^2 = 2k^2, so q^2 is even, and it follows that q is even. However, if both p and q are even, then they share a common factor of 2, and p/q is not reduced as much as possible. This is a contradiction to our original assumption, so it must be the case that the square root of 2 is not rational.

Therefore, the square root of 2 is irrational.In the above proof, each step is illustrated. For step one, we assumed the opposite of the statement ‘the square root of 2 is irrational’, so we assumed that the square root of 2 is rational.

For step two, we proceeded to try to prove that the square root of 2 is rational. Step three was to find a contradiction, and we found that when we saw that p/q was found to be both reducible and irreducible. This led us to step four, where we stated that our assumption, that the square root of 2 was rational can’t be true, so the square root of 2 must be irrational.Let’s consider another simple example.

Assume we want to prove the following statement.Statement: 193 is an odd number.While we could prove this directly by showing that 193 = 2 * k + 1 where k is an integer, we will use proof by contradiction to further illustrate this method.Proof by contradiction:Assume not.

That is, assume that 193 is not an odd number. Since 193 is an integer and by assumption it is not an odd number, it must be an even number. Then there exists an integer s such that 193 = 2 * s.

193 is an odd number

However, if s = 96.5, it is not an integer, so we have a contradiction.

Therefore, it can’t be true that 193 is an even number, so 193 must be an odd number.

Lesson Summary

We now have another method of proof in our toolbox, namely proof by contradiction – one of our ‘finest weapons’ when it comes to proving a statement or argument. To carry out proof by contradiction, first ask yourself what would happen if your statement weren’t true. If you realize that would lead to a contradiction, then proof by contradiction is a good way to go. After you have decided this is the method you want to use, follow the steps outlined in this lesson to successfully prove your statement.

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