Answer:
The relation [tex]\texttt{r}[/tex] on the set of integers, where [tex]a\, \texttt{r}\, b[/tex] if and only if the product [tex]a\, b[/tex] is even, is not reflexive. Counterexample: it is not true that [tex]1\, \texttt{r}\, 1[/tex] ([tex]a = b = 1[/tex]) since [tex]1\times 1 = 1[/tex] is not an even number.
Step-by-step explanation:
A relation [tex]\texttt{r}[/tex] on a set is reflexive if and only if for any element [tex]x[/tex] in that set, [tex]x\, \texttt{r}\, x[/tex].
Verifying whether the relation [tex]\texttt{r}[/tex] in this question is reflexive is equivalent to verify whether [tex]x\, \texttt{r}\, x[/tex] for all integer [tex]x[/tex]- in other words, whether the product [tex](x)\, (x) = x^{2}[/tex] is an even number.
Since not all perfect squares are even numbers, this statement would not be true. For example, for [tex]x = 1[/tex], [tex]x^{2} = 1[/tex] isn't an even number. Hence, the relation [tex]\texttt{r}[/tex] is not reflexive.
