Here's how it works:
1. Assume the opposite: You begin by assuming that the statement you want to prove is *false*.
2. Logic and deduction: You then use logical reasoning and deductions based on this assumption.
3. Reach a contradiction: The goal is to reach a contradiction, a statement that is clearly false or illogical.
4. Conclude the original statement is true: Since the assumption led to a contradiction, you conclude that the original assumption must be false. This means the statement you wanted to prove must be true.
Example:
Prove that the square root of 2 is irrational.
1. Assume the opposite: Assume that the square root of 2 is rational. This means it can be written as a fraction a/b, where a and b are integers and b is not zero.
2. Logic and deduction: If the square root of 2 is a/b, then squaring both sides gives us 2 = a^2/b^2. This implies that a^2 = 2b^2.
3. Reach a contradiction: Since a^2 is even (because it's equal to 2 times another number), a must also be even. This means a can be written as 2k, where k is another integer. Substituting this into the equation a^2 = 2b^2, we get (2k)^2 = 2b^2, which simplifies to 4k^2 = 2b^2. Dividing both sides by 2 gives us 2k^2 = b^2. This means b^2 is also even, and therefore b must be even.
4. Conclude the original statement is true: We've shown that both a and b are even if the square root of 2 is rational. However, this contradicts our original assumption that a/b was in its simplest form, meaning a and b have no common factors. Therefore, the assumption that the square root of 2 is rational must be false. This means the square root of 2 is irrational.