$(10$ points$)$ Propositional Satisfiability

Determine whether the following compound propositions are satisfiable.

$($a$)($p $$q$)($p $->$ q$)$

$($b$)($p $->$ q$)($q $->$p$)($p $$ q$)$

Solution:

$6.(10$ points$)$ Quantifiers

Suppose P $($x$,$ y$)$ means $$x $+2$y $=$ xy$,$ where x and y are integers. Determine the truth value of the

statement.

$($a$)$x$$y P $($x$,$ y$)$

$($b$)$x$$y P $($x$,$ y$)$

$($c$)$y$$x P $($x$,$ y$)$

$($d$)$y$$x P $($x$,$ y$)$

$($e$)$x$$y $$P $($x$,$ y$)$

Solution:

$7.(10$ points$)$ Negation

Express the negation of the statement in terms of quantifiers without using the negation symbol.

$($a$)$x$(($x $>1)($x $<1))$

$($b$)$x$(3<$ x $<=7)$

Solution:

$4$