A is the father of X and B is the mother of Y. Y is the sister of both X and Z. Which of the following statements is definitely not true?

Introduction / Context:
This problem mixes blood relationships and logical certainty. You know that Y is a sister, and you are asked to decide which given statement is definitely not true, meaning it can never be correct under any valid family arrangement. You must distinguish between statements that might be true in some possible family structure and one statement that directly contradicts the information given in the question.

Given Data / Assumptions:
- A is the father of X. - B is the mother of Y. - Y is the sister of X and also the sister of Z. - Sister means Y is female. - We do not initially know the genders of X and Z. - We are to identify which option can never be true, no matter how we complete the family tree.

Concept / Approach:
The word definitely in such questions is crucial. A statement is definitely not true if it logically contradicts the basic facts provided. On the other hand, a statement is possibly true if you can construct at least one consistent family tree in which it holds. The most reliable strategy is to use the gender information of Y and track possible parent pairs, then test each option one by one against this framework.

Step-by-Step Solution:
Step 1: From Y is the sister of X and Z, we know Y is a female child who shares at least one parent with X and Z. Step 2: A is the father of X, and B is the mother of Y. A reasonable and common exam assumption is that A and B form a couple and that X, Y and Z are their children, but even if we allow half-siblings, Y being a sister remains female. Step 3: Examine option (c): Y is the son of A. This explicitly describes Y as a son, which means a male child. Step 4: However, the stem clearly states that Y is a sister of X and Z. The term sister unambiguously identifies Y as female, not male. Step 5: Therefore, describing Y as a son conflicts directly with the given data; no consistent family structure can make Y both sister (female) and son (male) at the same time. Step 6: All other options can be made true in at least one plausible arrangement: B can be Z's mother, X may be either brother or sister of Z, and B may be A's wife in a typical family configuration.

Verification / Alternative check:
Try to sketch a family tree where option (c) is true. You would need Y to be male to be a son, but the word sister forces Y to be female. No amount of rearranging or adding relatives removes this contradiction. In contrast, it is easy to build a tree where A and B are married, X, Y and Z are their children, Y is a daughter, and then B is the mother of Z and X is either a sister or brother of Z depending on X's gender. This confirms that only option (c) is impossible.

Why Other Options Are Wrong as the Answer:
- B is the mother of Z: This is entirely possible if X, Y and Z are all children of A and B. - X is the sister of Z: This can be true if X is female; the stem does not specify X's gender. - B is the wife of A: This is a natural pairing if A and B are the parents of the siblings. - X is the brother of Y: Also possible if X is male and they share parents.

Common Pitfalls:
A common mistake is to think that definitely not true means not necessarily always true. That is incorrect; here it means logically impossible. Another pitfall is ignoring the gender encoded in the word sister and treating it like a neutral sibling. Always pay close attention to gender-specific terms in blood relation questions.

Final Answer:
The only statement that can never be true is Y is the son of A.