Let's analyze both the logics A and B.
In A, it says there exists an x such that both P(x) AND Q(x) are true. While in B, it says that for some x, P(x) is true, and for some y (y can be same as x, or anything other than x) Q(y) is true.
Let's take an example.
Suppose P(x) means x is a boy. And Q(x) means x is smart. And the universe of discourse be the set of all students in a class.
Now, A here means, that there is a student who is a boy AND is smart. And B means, there is a student who is a boy AND there is a student who is smart.
Now, if Rishabh is a boy and he is smart than A is true, and of course, B is also true, because there is a student, Rishabh who is a boy AND there is a student Rishabh who is smart.
While suppose there is no boy in the class who is smart, all are dumb. Then A is not true, while B still may be true. B will be true when there is at least one boy in the class and at least one girl who is smart.
So, this tells that B implies A is not true as well as the double implication is also not true.
Now, A implies B can be seen true intuitively from the above example. More clearly you can see that A is true whenever there is an element for which both are true. So, for the same element, they (P and Q) both individually will also be true. So, whenever A will be true, then B will definitely be true.
So, the correct answer is a) A=> B
Hope this makes sense :)
