Question

determine whether the statement is true or false. p → q can be translated as \p is necessary for q.\ a. the statement is true because p → q means \if p, then q,\ so q cannot occur without p. b. the statement is false because p → q means \p is sufficient for q\ so q can still occur even if p does not, but if p occurs, so does q. c. the statement is true because p → q means \q is sufficient for p,\ which is the same as saying that p is necessary for q. d. the statement is false because p → q means \q is necessary for p,\ not the other way around.

Explanation:

The conditional statement \(p
ightarrow q\) is read as "if \(p\), then \(q\)", which means \(p\) is sufficient for \(q\). That is, whenever \(p\) occurs, \(q\) occurs, but \(q\) can occur even when \(p\) does not. The statement " \(p\) is necessary for \(q\)" would be translated as \(q
ightarrow p\). So the given translation of \(p
ightarrow q\) as " \(p\) is necessary for \(q\)" is false.

Answer:

B. The statement is false because \(p
ightarrow q\) means " \(p\) is sufficient for \(q\)" so \(q\) can still occur even if \(p\) does not, but if \(p\) occurs, so does \(q\).