Every student in this class has studied calculus
If S(x) represents the statement that person x is in this class, C(x) which is the statement “x has studied calculus.”
we see that our statement can beexpressed as

∀x(S(x) → C(x) )

Some student in this class has visited Mexico

M(x), which is the statement “x has visited Mexico.” S(x) to represent “x is a student in this class.”

∃x(S(x) ∧ M(x))

In 1st example Implication is taken Why in 2nd Example and is taken

For the second Statement ∃x(S(x) ∧ M(x)) the student has to be in the class and has visited Mexico. Since ∃x is used, the result turns out to be true even if one student has visited Mexico. Suppose the student is not in the class then also the result turns out to be true if ∃x (S(x) ->M(x)).

Think Logically by getting the basic definition of ∃ and ∀, ∃ is actually Infinite 'OR' i.e p1 or p2 or p3 or p4... and so on whereas ∀ is actually infinite 'AND' . Now if suppose your domain is students in your class and if you use ∀x(S(x) → C(x) ) this makes sure that every student outside the domain is evaluated to True(you know about truth table of implication if the student is not in domain then s(x) is false and hence s(x)->c(x) is evaluated to True). Now suppose you take the same ∃x(S(x) → M(x)) if student is not in domain the result turns out to be true and because of that the whole expression turns out to be true which we do not want we only want to check those students which are in the domain and hence ∃x(S(x) ^M(x)) is the right way according to the specification.

If you will replace ^ symbol with -> in 2nd example, you will found out like ∃x(S(x) ->M(x)) which means there exist a student,if student belong to the class then it must have visited Mexico, this is wrong actually , we should go like this, "There exist a student(some student /at least one or more) who belong to the class and he has visited Mexico" so it will go like this as ∃x(S(x) ∧ M(x))

in second statement x refers to domain of only students it is true only for domain of students however for first domain is not specified so it is true for those who are not student of the class i hope this clearly demarks the difference between the two