Consider the following statements(a) If D is dog, then D is very good.(b) If D is very good, then D is dog.(c) If D is not very good, then D is not a dog.(d) If D is not a dog, then D is not very good.Identify the pairs of statements having the same…

Answer: (a) and (c) have the same meaning; (b) and (d) have the same meaning.

Let \(p\): D is dog, \(q\): D is very good. Then (a) is \(p \rightarrow q\), (b) is \(q \rightarrow p\) (converse), (c) is \(\sim q \rightarrow \sim p\) (contrapositive) and (d) is \(\sim p \rightarrow \sim q\) (inverse).

The columns of (a) and (c) are identical, so \(p \rightarrow q \equiv \sim q \rightarrow \sim p\): a conditional and its contrapositive are logically equivalent. The columns of (b) and (d) are identical, so \(q \rightarrow p \equiv \sim p \rightarrow \sim q\): the converse and the inverse are equivalent. Hence the pairs with the same meaning are (a), (c) and (b), (d).