# Chapter 3 - Logic - 3.3 Truth Tables for Negation, Conjunction and Disjunction - Exercise Set 3.3: 31

$\begin{array}{llllll} p & q & \sim p & \sim q & p\wedge\sim q & \sim p\vee( p\wedge\sim q)\\ \hline T & T & F & F & F & F\\ T & F & F & T & T & T\\ F & T & T & F & F & T\\ F & F & T & T & F & T \end{array}$

#### Work Step by Step

Set up a truth table for two inputs, p and q: $\begin{array}{llll} p & q & ... & ...\\ \hline T & T & & \\ T & F & & \\ F & T & & \\ F & F & & \end{array}$ In columns 3 and 4, use the negation table for $\sim p$ and $\sim q$ $\begin{array}{llllll} p & q & \sim p & \sim q & & \\ \hline T & T & F & F & & \\ T & F & F & T & & \\ F & T & T & F & & \\ F & F & T & T & & \end{array}$ In the next column (5th) use inputs from columns 1 and 4 for $\sim p\wedge\sim q$ and the conjunction truth table, (true only when both inputs are true) $\begin{array}{llllll} p & q & \sim p & \sim q & p\wedge\sim q & \\ \hline T & T & F & F & F & \\ T & F & F & T & T & \\ F & T & T & F & F & \\ F & F & T & T & F & \end{array}$ Finally, for $\sim p\vee( p\wedge\sim q)$, use disjunction truth table on inputs from columns 3 and 5 (true if at least one input is true) $\begin{array}{llllll} p & q & \sim p & \sim q & p\wedge\sim q & \sim p\vee( p\wedge\sim q)\\ \hline T & T & F & F & F & F\\ T & F & F & T & T & T\\ F & T & T & F & F & T\\ F & F & T & T & F & T \end{array}$

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