$5.(5$ marks$)$ Consider the following predicates and logical statements:

P $($x$)$ : x $=2$k $+1$ for some k in Z S$($x$,$ y$)$ : x $<=$y$1$

$($a$)$x in Z $$y in Z such that S$($x$,$ y$)$

$($b$)$x in N such that P $($x$)$