Equivalences

Some rules (and tips) on building truth tables for logic expressions:

• The few leftmost columns of your table should be the simple variables the expression contains. For example, if your expression has the variables $p$, $q$, and $r$ in it, your table should start with 3 columns, one for each of $p$, $q$, and $r$.
• The number of rows that your table will have can be calculated as follows: $2^{(\text{amount of variables})}$. For example, if your expression has the variables $p$, $q$, and $r$ in it, the truth table will feature $2^3 = 8$ rows of bits.
• To make the evaluation of expressions gradual and easy, include additional columns in your table that correspond to intermediate expressions that build up the final expression. For example, if your expression is $\neg(p \vee q)$, don't include just the columns $p$, $q$, and then right away $\neg(p \vee q)$. Instead, include also a column for the $\vee$ operation: $p \vee q$, which is an intermediate operation that you perform before applying the negation operation.
• The 'variables' $1$ and $0$ are special: while variables like $p$ and $q$ vary (each of them can be either $0$ or $1$,) $1$ and $0$ are actually constants. If you need to include a column for $1$, all the bits across this column will be $1$s. Similarly, if you need to include a column in the truth table for $0$, all the bits across this column will be $0$s.