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[Compound Truth Table] A 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[Role of Parentheses] 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. ::: When building a truth table, write one column for each important substatement. For $q\land(\neg r\to p)$, the smaller parts are $\neg r$, then $\neg r\to p$, and finally $q\land(\neg r\to p)$. This method prevents mistakes because each column uses values already computed in earlier columns. :::example[Truth Table for a Compound Statement] Let $$\begin{align*} p &: \text{The device is charged.}\\ q &: \text{The device is switched on.}\\ r &: \text{The network is unavailable.} \end{align*}$$ Consider the compound statement $q\land(\neg r\to p)$. This statement says that the device is switched on and, if the network is available, then the device is charged. ::: :::definition[Truth Table for a Grouped Conditional] The truth table for $q\land(\neg r\to p)$ is $$\begin{equation*} \begin{array}{c|c|c|c|c|c} p & q & r & \neg r & \neg r\to p & q\land(\neg r\to p)\\ \hline 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 & 0 & 0\\ 0 & 1 & 1 & 0 & 1 & 1\\ 1 & 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 1 & 0\\ 1 & 1 & 0 & 1 & 1 & 1\\ 1 & 1 & 1 & 0 & 1 & 1 \end{array} \end{equation*}$$ Given that the final statement is a conjunction, the last column is true only when both $q$ and $\neg r\to p$ are true. ::: :::example[Parentheses in Compound Statements] Compare $p\lor(q\land r)$ and $(p\lor q)\land r$. The first statement performs the conjunction $q\land r$ before the disjunction. The second statement performs the disjunction $p\lor q$ before the conjunction with $r$. Hence, the two statements may have different truth values. ::: Use the comparator to test two compound statements made from the same primitive statements. Choose values for $p$, $q$, and $r$, then compare $p\lor(q\land r)$ with $(p\lor q)\land r$. The two expressions can agree in some rows and disagree in others. This shows why parentheses are part of the logical structure, not just a typographical detail. :::scientific-preview[Parentheses Truth Table Comparator] ::: :::definition[Truth Tables Showing the Role of Parentheses] The comparison table is $$\begin{equation*} \begin{array}{c|c|c|c|c|c|c} p & q & r & q\land r & p\lor(q\land r) & p\lor q & (p\lor q)\land r\\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 0 & 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 1 & 1 & 1\\ 1 & 1 & 0 & 0 & 1 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array} \end{equation*}$$ The final columns differ in some rows. Therefore, the two compound statements are not identical. ::: :::solved-problem Construct the truth table for $\neg(p\land q)\to\neg q$. ::: :::solution Let the intermediate columns be $p\land q$, $\neg(p\land q)$, and $\neg q$. Then $$\begin{equation*} \begin{array}{c|c|c|c|c|c} p & q & p\land q & \neg(p\land q) & \neg q & \neg(p\land q)\to\neg q\\ \hline 0 & 0 & 0 & 1 & 1 & 1\\ 0 & 1 & 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 1 & 1 & 1\\ 1 & 1 & 1 & 0 & 0 & 1 \end{array} \end{equation*}$$ Hence, the compound statement is false only when $p=0$ and $q=1$. ::: :::exercise Construct truth tables for: (1) $p\to(r\to q)$. (2) $(q\to p)\to r$. (3) $(r\to q)\to(q\to r)$. (4) $[q\land(q\to r)]\to r$. ::: :::answer (1) Use intermediate column $r\to q$, then $p\to(r\to q)$. (2) Use $q\to p$, then $(q\to p)\to r$. (3) Use $r\to q$ and $q\to r$, then compare by implication. (4) Use $q\to r$, then $q\land(q\to r)$, then the final implication. ::: :::faq[Frequently Asked Questions] Q: Why do we need intermediate columns? A: They break a complex statement into smaller statements so each truth value is computed in the correct order. Q: Can different parentheses change a truth table? A: Yes. The statements $p\lor(q\land r)$ and $(p\lor q)\land r$ have different final columns. Q: Should the final column always be computed last? A: Yes. The final column depends on the intermediate columns. ::: :::call-to-action[Build Compound Tables Carefully] subtitle: Practise grouping with parentheses before using truth tables to identify tautologies and valid arguments. button: Next Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/tautologies-contradictions-and-valid-arguments button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/conditional-statements-and-programming-decisions :::