M and N had four children P, Q, R and S. Of them, only P and R were married. They had children X and Y respectively. If Y is a legitimate child of W, which one of the following statement is necessarily FALSE?

To determine which statement is necessarily FALSE, let us break down the family relations step-by-step:

1. Parents and Children:
M and N are partners/parents who have four children: P, Q, R, and S.

2. Marital Status:
Of these four children, only P and R were married. Q and S are unmarried.

3. Children of P and R:
P has a child named X, and R has a child named Y.

4. Parentage of Y:
We are told that Y is the child of R (since the problem states P and R had children X and Y respectively). We are also given that Y is a legitimate child of W. For Y to be a legitimate child of both R and W, R and W must be married to each other. Therefore, W is the spouse (wife or husband) of R.

5. Analyzing the Options:
- W is the wife of R: This is consistent with R and W being the parents of Y.
- R is the father of Y: This is consistent because Y is the child of R.
- W is the wife of P: Since W is married to R (as they share the child Y), W cannot be married to P. The problem explicitly states that only P and R were married, and Y is the legitimate child of W and R. Therefore, W cannot be the wife of P. This statement is necessarily FALSE.