Equivalences

• If you need to check whether two expressions are equivalent, you'll need to build 2 tables, one for each expression. You may, if wanted, 'glue' these tables to one another to create a single, longer table.
• Two expressions are equivalent if and only if the sequences of digits underneath the columns for each of these expressions are identical. In the example for $p \to q$ and $\neg q \to \neg p$, the sequence of digits for each of these was $(1, 1, 0, 1)$.
• If the sequences of digits differ from one another, even in just one of the digits, the two expressions are not equivalent.

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Let's verify a couple more identities. The truth table for the equivalence $p \wedge1 \Leftrightarrow p$ is:

$1$	$\boldsymbol p$	$\boldsymbol{p \wedge 1}$
$1$	$0$	$0$
$1$	$1$	$1$

Since the digits under $p$ and under $p \wedge 1$ are the same (i.e., $(0, 1)$), we successfully confirmed the equivalence.