| 单词 | proof by contradiction |
| 释义 | proof by contradiction A proof which works by showing that assuming that a proposition is false leads to a contradiction. That is, assuming the original proposition is false allows one to prove both some statement and its opposite: thus the initial assumption that the proposition was false cannot have been correct. For example, consider the proposition that the square root of 2 is not a rational number. Assume the contrary, that √2 is a rational number, x/y. If both x and y are even, divide both by 2, and continue until left with a rational x/y where at least one is odd. Now the square of any even number is even, and the square of any odd number is odd. x2 = 2y2 so x must be even. Set x = 2z. Then 4z2 = 2y2, so y2 = 2z2, so y must be even. But at least one of x and y was by construction odd. This shows that the assumption that √2 is a rational number produces a contradiction, so √2 cannot be rational. |
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