How to convert a truth table to boolean expression?

Question

If I have a huge truth table, it's hard for me to construct an expression. I know a problematic method, the Disjunctive Normal Form. But I found that I cannot reduce the huge expression.

\begin{array}{c | c | c | c || c} \hline p & q & r & s & \phi \\ \hline 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \hline \end{array}

Answer 1

Hint: when trying to simplify the disjunctive normal form for this expression, it may be helpful to use the absorption law. That is, for propositions $a$ and $b$:

$a \vee (a\wedge b) \equiv a$.

The intuition behind this law is fairly obvious, if $a$ and $b$ has a truth value, then of course $a$ alone has the same truth value. So there would be a redundancy in saying the left side when it suffices to say just $a$.

You may try to apply this same reasoning to something like $a \vee (a\wedge b \wedge c)$ as well.

Answer 2

Hint: Whenever $p$ and $r$ are both true ($1$), the expression is false ($0$). Can you say something similar about $q$ and $s$.

Answer 3

You should look up Karnaugh maps

With karnaugh maps you can turn any truth table into a minimally complex expression without too much brain power. I would have put this in a comment but I can't comment yet.