Question $4$: $(20$ points$)$

An argument consists of a list of premises and a conclusion. Consider the following list of premises, and the two possible conclusions, $($i$)$ and $($ii$).$

$$All students who are good at counting are also good at probability.$$

$$All students who are good at Q$-$logic are also good at programming.$$

$$Some students are good at counting.$$

$$Some students are good at Q$-$logic.$$

$($a$)(8$ points$)$ Represent these premises in inference format in the language of quantified logic. Define $$x$$ to be a student and carefully define your open statements. $($There are four open statements to define.$)$

$($b$)(12$ points$)$ For each of the following two possible conclusions,

$($i$)$Some students are good at probability, and some students are good at programming.$$

$($ii$)$Some students are good at both probability and programming.$$

IF the argument is logically valid, then prove it using inference formalism.

IF the argument is logically invalid, then demonstrate a counterexample.

$($Hint: Consider carefully the two possible conclusions $$ they do not have the same meaning.$)$

using discrete mathematics