Given, $T (x, y)$ mean that student $x$ likes cousin $y$, where the domain for $x$ consists of all students at your school and
the domain for $y$ consists of all cousins.
$(a) : $ $\exists x \exists z \forall y(T(x,y) \leftrightarrow T(z,y))$
See, $\exists x \exists z $, means that there is some(at least one) Pair $(x,z)$ of students, for which the rest of the formula is True.
So, You can say that there is some(at least one) Pair $(x,z)$ of students such that $x$ likes cousin $y$ if and only if $z$ likes cousin $y$. i.e. For All Cousins $y$, either $x,z$ both like him or none of $x,z$ like him.
So, Final Interpretation : There is at least one student $x$, for whom there is at least one student $z$ such that for all cousins $y$-----, $x,z$ agrees on liking him.
Or, There is some pair of Students $(x,y)$ such that for all cousins, $x$ likes a cousin if and only if $y$ likes a cousin.
Or, There are two students(not necessarily Distinct) at your school who like exactly the same set of cuisines.
Now some Notable Points about it :
1. As long as Domain of Students is Non-empty, the given Formula is ALWAYS True. because It is Always True for every Student individually. i.e. Say, Simmi is a Student then for the pair $(simmi,simmi)$, this formula is trivially True.
2. There need not be Two different students to make it True.... One student is Enough to make it True.
So, Conclusion is that $\exists x \exists z \forall y(T(x,y) \leftrightarrow T(z,y))$ is ALWAYS True for Non-empty Student Domain.
$(b) : $ $\forall x \forall z \exists y(T(x,y) \leftrightarrow T(z,y))$ :
For all pairs $(x,z)$ of students, there is at least one cousin $y$ such that both $x,z$ agrees on him. Or, For every pair of students at your school, there is some cousin about which they have the same opinion (either they both like it or they both do not like it).
what is the difference between both of them?
Lots of difference.
This is like this :
Interpretation of $\forall x \exists y S(x,y)$ : Have you ever heard Raabta Song which goes like this---- "Kehte Hain Khuda Ne Iss Jahan Mein Sabhi Ke Liye Kisi Na Kisi Ko Hai Banaya Har Kisi Ke Liye" ... Yeah..It's the exact interpretation of $\forall x \exists y S(x,y)$.. For every element $x$, there is some element $y$($y$ could be different for different $x$ but for each $x$ there must someone), such that Property(predicate) $S(x,y)$ is True.
Interpretation of $\exists y \forall x S(x,y)$ : Have you watched "Mahabharata" ? ..If you have, then remember Draupdi...One wife for All five pandavas.. Yes, It's the exact interpretation of $\exists y \forall x S(x,y)$.. One for All.
https://gateoverflow.in/168951/predicate-logic?state=edit-219597
