Thanks bro for answering, your answer is correct, although i have one doubt. Why S1 is not true.
for S1 we can write like this - for every x there exist a number y in which the x can divide y. what i understand for every number in the integer it can divide at least one number. it must be true.

bro i think it means for every value of x there exists at least one y. The value of y depends on what we choose x. suppose x = {1,2,3,4} It means for every value of x i.e if we take x =1 it should divide one number (as it divide 1) and if we take x =2 it should also divide one number and so on. So every value of x, it should divide some number and it is not necessary that some number should be the same for different x.

like everyone like someone it means that someone should not be the same for everyone.

this is the way i am thinking, Please correct me if i am wrong.

see the first thing you need to understand is that..

universe of discourse is the set of all integers

and the statment means that every x should be divisor of y..

and here set x will contain entire integers which is countably infinite..

now say if you want to satisfy statement 1 than there is only 1 y which can be divided by all values of x i.e. product of all integers from the set x... so now tell me can u find that value of y...

i do understand the universe is set of all integers,

what you are doing wrong here is you are finding one y for all x which is wrong, you have to find at least one y for each and every x. there is a difference between

∀x ∃y P(x,y) and ∃y ∀x P(x,y)

The reason why S1 is false because for x=0 we can't find at least one y if the question would have been ∀x ∃y P(x,y) where x is not 0. Then it would be true.

yes i got u i was making a mistake i was taking a different path which was wrong and invalid. it means that all these values of x must divide some y or atleast one y..