Exercise 17 Page 111

To write the converse, inverse, and contrapositive of a given statement, we first need to rewrite it in if-then form. We have to identify the statement's hypothesis and conclusion. In the if-then form, the hypothesis comes after if and the conclusion comes after then. The given statement says that All whole numbers are integers. Here, we start with having all whole numbers and conclude that they are integers. With this information, we can identify the $\text{hypothesis}$ and the $\text{conclusion}.$ Now we can write the statement in if-then form. Now let's consider how the $\text{hypothesis}$ and the $\text{conclusion}$ will relate to one another in the different statements symbolically.

Related Conditionals	Symbols
Given statement	$p\to q$
Converse	$q\to p$
Inverse	$\neg p\to \neg q$
Contrapositive	$\neg q\to \neg p$

Finally, we can use the symbolic statements to write the verbal statements.

Related Conditionals	Symbols	Statement	Truth Value	Counterexample
Given statement	$p\to q$	If $\text{a}$ $\text{number is}$ $\text{a}$ $\text{whole}$ $\text{number},$ then $\text{it}$ $\text{is}$ $\text{an}$ $\text{integer}.$	True	-
Converse	$q\to p$	If $\text{a}$ $\text{number}$ $\text{is}$ $\text{an integer,}$ then $\text{it}$ $\text{is}$ $\text{a whole number}.$	False	$\text{-}3$ is an integer but it is not a whole number.
Inverse	$\neg p\to \neg q$	If $\text{a}$ $\text{number}$ $\text{is}$ $\text{not}$ $\text{a whole number},$ then $\text{it}$ $\text{is}$ $\text{not}$ $\text{an integer}.$	False	$\text{-}3$ is not a whole number but it is an integer.
Contrapositive	$\neg q\to \neg p$	If $\text{a}$ $\text{number}$ $\text{is}$ $\text{not}$ $\text{an integer},$ then $\text{it}$ $\text{is}$ $\text{not}$ $\text{a whole number}.$	True	-