Exercise 35 Page 105

In order to make the truth table, we will make columns that show the truth value of $p,q,r,\neg q,(\neg q\vee r)$ and $p\wedge (\neg q\vee r).$ Thus, we can 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 negation of statement $q.$ 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.

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

Let's determine the truth value of Statement $p\wedge (\neg q\vee r)$ given that Statement $p$ and Statement $r$ are true. We have two cases that satisfy the given condition.

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

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