Equivalence in a Formal Mathematical Proof

Consider the following statement:

If, and only if, the door is open, then the refrigerator light is on.

In this example, we have two propositions

\begin{aligned} D &= \text{The door is open} \\ L &= \text{The refrigerator light is on} \\ \end{aligned}

This is an example of an equivalence statement, which again relates two propositions: a hypothesis and a conclusion. An equivalence statement tells us that the hypothesis and conclusion are either both true, or both false. We write the equivalence statement more concisely as

$D \Leftrightarrow L$

Similarly to an implication statement, we can summarise an equivalence statement in a truth table

If the light is on, then the statement is only true if the door is open
- If the door is open, then the equivalence holds and is therefore true
- If the door is closed, then the light is not on 'only if' the door is open, so the equivalence is false.
If the light is off, then the statement is only true if the door is closed
- If the door is open, then the light is not on 'if' the door is open, so the equivalence is false.
- If the door is closed, then the equivalence holds and is therefore true

$D$	$L$	$D \Leftrightarrow L$
True	True	True
True	False	False
False	True	False
False	False	True

We say that an equivalence statement is stronger than an implication statement, because it makes more statements about the world. An implication statement tells us that the conclusion is true when the hypothesis is true, but does not tell us anything about the world when the hypothesis is false. An equivalence statement goes further: it tells us that the conclusion is true when the hypothesis is true like an implication statement, but it also tells us that the conclusion is false when the hypothesis is false.

...