Explain, with the aid of an example, what is meant by proof by contradiction - Leaving Cert Mathematics - Question 6A - 2012

Consider the statement The lines $y = x$ and $y = x + 1$ do not intersect.

We can prove this by contradiction as follows. Assume that the statement is false. So there is some point $(a, b)$ that lies on both lines. Therefore, $b = a$ since the point is on the line $y = x$ and $b = a + 1$ since the point is on the line $y = x + 1$.

Combining these equations, we get $a = a + 1$ which implies that $0 = 1$

However, this contradicts the fact that $0 eq 1$. Therefore, our original assumption is false. So we have proved that the lines $y = x$ and $y = x + 1$ do not intersect.