Exercise 37 Page 105

In order to make the truth table, we will make columns that show the truth value of $p,q,r,\neg q,\neg r,(\neg p\vee q)$ and $(\neg p\vee q)\wedge r.$ Thus, we will be able to examine the main statement in two cases using one truth table. Notice that we have both conjunction with the sign "$\wedge$" and disjunction with the sign "$\vee .$" We also need to write the negations of statement $q$ and statement $r.$ Therefore, we need to remember three things.

• A conjunction is only true when both statements that form it are true.
• A disjunction is true if at least one of the statements is true.
• The negation of statement has the opposite meaning, as well as an opposite truth value.

Now, we are ready to construct the table.

$p$	$q$	$r$	$\neg q$	$\neg r$	$(\neg q\wedge \neg r)$	$p\vee (\neg q\wedge \neg r)$
T	T	T	F	F	F	T
T	T	F	F	T	F	T
T	F	T	T	F	F	T
T	F	F	T	T	T	T
F	T	T	F	F	F	F
F	T	F	F	T	F	F
F	F	T	T	F	F	F
F	F	F	T	T	T	T

Let's determine the truth value of statement $(\neg p\vee q)\wedge r$ given that statement $p,$ statement $q,$ and statement $r$ are true. We have one case that satisfies the given condition.

$p$	$q$	$r$	$\neg q$	$\neg r$	$(\neg q\wedge \neg r)$	$(\neg p\vee q)\wedge r$
T	T	T	F	F	F	T

As we can see, in this case, $(\neg p\vee q)\wedge r$ is true.