1. If there exist x for which P is true then for all such x Q will be true.
Hence is P is true for all x then Q is true for all x (Correct)
2. there exist x for which either of P or Q is true. Then if for all x if P is true then for all x Q is true.
Not Necessary : consider a case when Q is false for all x , and P is true for all x.
3. here operator is <====> , we can consider any of LHS or RHS first.
lets consider RHS first If there some x for which P is true and there is some other value of x (Note we are not using same value of x for both) for which Q is true . Then it is not necessary that there exist common value of x for which P and Q both are true.
4.
For every value in X there exist some value in Y to which it is related to.
But there does not exist any value for Y which is related to all values of X