Parentheses in Compound Truth Tables

GROUPING COMPOUND STATEMENTS

Conditional program decisions often contain several connectives at once. Parentheses in compound truth tables tell us which part must be evaluated first. Without parentheses, the meaning of a symbolic statement may become ambiguous or may be read incorrectly. In this lesson, parentheses in compound truth tables will be used to construct intermediate columns in the correct order .

DEFINITION

A \textbf{compound truth table} is a truth table constructed for a statement that contains several logical connectives. Such a truth table is built by forming intermediate columns for smaller parts of the compound statement.

DEFINITION

\textbf{Parentheses} determine the order in which logical connectives are applied. Different placements of parentheses may produce different compound statements. Thus, parentheses are necessary when a statement can be grouped in more than one way.

q\land(\neg r\to p)

EXAMPLE

\begin{align*} p &: \text{The device is charged.}\\ q &: \text{The device is switched on.}\\ r &: \text{The network is unavailable.} \end{align*}

DEFINITION

q\land(\neg r\to p)

EXAMPLE

p\lor(q\land r)

p

Dynamic Sandbox

Initializing Workspace

DEFINITION

\begin{equation*} \begin{array}{c|c|c|c|c|c|c} p & q & r & q\land r & p\lor(q\land r) & p\lor q & (p\lor q)\land r\\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 0 & 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 1 & 1 & 1\\ 1 & 1 & 0 & 0 & 1 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array} \end{equation*}

SOLVED PROBLEM

\neg(p\land q)\to\neg q

SOLUTION

p\land q

EXERCISE

p\to(r\to q)

ANSWER

r\to q

3 Questions

They break a complex statement into smaller statements so each truth value is computed in the correct order.

p\lor(q\land r)

Yes. The final column depends on the intermediate columns.

Practise grouping with parentheses before using truth tables to identify tautologies and valid arguments.