A calculus of the absurd

5.2 Proving identities

When proving identities (i.e. expressions in the form \(A = B\)) there are two ways to do it.

  • • Pick a side (any side, but as a rule of thumb the one which looks more complex/messy) and show that there exist a sequence of steps (applications of statements that one knows to be true, e.g. that \(-1 \leqq \sin (x) \leqq 1\) or if \(a \ne 0\) and \(ab = ac\), then \(b = c\)) which rewrite one side as the other. Important: these steps must be invertible (i.e. they can be applied both ways, as otherwise the equality does not run in both directions - for example, squaring can be dangerous when working from one side to the other, as when taking the square root there are two solutions) as this is the essence of the “trick” which allows us to save half the work.

  • • Show that we can rewrite \(A\) and \(B\) and \(B\) as \(A\). The only advantage of this method over the previous one is that we may use irreversible steps.

There is also another way to prove that \(A = B\) which relies on the fact that the \(\leqq \) operator is “antisymmetric”, which is maths jargon for \(A = B\) if and only if \(A \leqq B\) and \(B \leqq A\). This technique can be really powerful in many situations.

TODO: example of using antisymmetric property of \(\leqq \) to prove things