Family Tree Deduction

We need to determine the family relationships based on the given information to find the statement that is necessarily FALSE.

Information Breakdown

• Parents: M and N
• Children of M & N: P, Q, R, S
• Married Children: P and R
• Children of Married Parents: P has child X; R has child Y.
• Child Y: Y is a legitimate child of W.

Relationship Analysis

• From "R has child Y" and "Y is a legitimate child of W", we deduce that R and W are the parents of Y.
• Since Y is a legitimate child, R and W are married.
• Given R is a male child of M and N (sibling to P), R is the father of Y.
• Therefore, W must be the mother of Y and the wife of R.
• M is the parent of R, making M the grandmother of Y.

Evaluating Statements

• Statement 1: M is the grandmother of Y. Since M is the parent of R, and R is the parent of Y, M is indeed the grandmother of Y. (TRUE)
• Statement 2: R is the father of Y. As deduced above, R and W are the parents of Y, and R is the father. (TRUE)
• Statement 3: W is the wife of R. Since Y is the legitimate child of R and W, and R is the father, W is the mother and thus the wife of R. (TRUE)
• Statement 4: W is the wife of P. P and R are siblings. W is the wife of R. Therefore, W cannot be the wife of P. (FALSE)

The statement that is necessarily FALSE is that W is the wife of P.