Answer
14a. See the truth table.
14b. Two logically equivalent ways to say the sentence, "If n is prime, then n is odd or n is 2:"
1. If n is prime and n is not odd, then n is 2.
2. If n is prime and n is not 2, then n is odd.
Work Step by Step
14a. To construct the truth table, first fill in the 8 possible combinations of truth values for p, q, and r. Then fill in the columns for ~q and ~r. Then fill in the truth values for q $\lor$ r according to the definition of OR (q $\lor$ r is true when either q is true, or r is true, or both q and r are true; it is false only when both q and r are false). Then fill in the truth values for (p $\land$ ~q) and (p $\land$ ~r) according to the definition for AND (a $\land$ b is true when, and only when, both a and b are true. If either a or b is false or if both are false, then a $\land$ b is false). Lastly evaluate the truth values for the three if/then statements. For an if/then statement, when the if element is T and then element is F, the statement is F. In all other cases the statement is T. Notice that the truth values for the three if/then statements are the same. Hence they are logically equivalent.
14b. The corresponding symbolic form for the statement, ""If n is prime, then n is odd or n is 2," is
p $\rightarrow$ q $\lor$ r.
The corresponding elements are:
p: n is prime
q: n is odd
r: n is 2
Hence,
p $\land$ ~q $\rightarrow$ r is "If n is prime and n is not odd, then n is 2."
p $\land$ ~r $\rightarrow$ q is "If n is prime and n is not 2, then n is odd."
