Let Loves $(b,g)$ denote that boy $b$ loves girl $g.$ If Sam loves Mary and Mary does not love Sam back,

then we say that "Sam loves in vain."

$($a$)$ Express the following statements using universal and existential quantifiers. Move any negations

to the right.

i$.$ "Sam has loved in vain."

ii$.$ "There is a boy who has loved in vain."

iii. "Every boy has loved in vain."

iv$."$No boy has loved in vain."

$($b$)$ For each of the above statements and each of the two relations below either prove that the

statement is true for the relation or that it is false.

Sam

Sam longrightarrow Mary

$($c$)$ Use Loves $(p_1,p_2,t)$ to add in the time at which it is occuring. Express the statement "Everyone

has loved in vain at some point."

$($d$)$ Sketch how the game would go to prove this statement.