Determine the truth value of each of the following propositions, when the

universe of discourse is either R

$+$ or Z $($consider each case separately$).$

$($a$)($x$)($x

$23=0)$ for Case $1$: x$,$ y in R and Case $2$: x$,$ y in Z$.$

$($b$)($x$)($x $<$ x $+\backslash$pi $)$ for Case $1$: x$,$ y in R and Case $2$: x$,$ y in Z$.$

$($c$)($x$)($y$)($xy $=1)$ for Case $1$: x$,$ y in Q and Case $2$: x$,$ y in Z$.$

$($d$)($y$)($x$)($xy $=1)$ for Case $1$: x$,$ y in Q and Case $2$: x$,$ y in Z