If 'A $ B' means 'A is the brother of B', 'A @ B' means 'A is the wife of B', 'A # B' means 'A is the daughter of B' and 'A * B' means 'A is the father of B', then which of the following expressions correctly indicates that U is the father-in-law of P?

Introduction / Context:
This question uses symbolic relationships to represent family ties. You are told what each symbol means and must choose the expression that shows U is the father-in-law of P. Father-in-law of P means that U must be the father of P's spouse. Such symbolic reasoning questions test your ability to decode a string of relations step by step and see the implied family structure.

Given Data / Assumptions:

• A $ B: A is the brother of B (A is male, sibling of B).
• A @ B: A is the wife of B (A is female, married to B).
• A # B: A is the daughter of B (A is female child of B).
• A * B: A is the father of B (A is male parent of B).
• We want an expression in which, after decoding, U turns out to be the father of P's spouse.

Concept / Approach:
To represent U as father-in-law of P, we must ensure two things: P is married to some person, and U is the father of that person. In symbolic form this means P must appear as someone's spouse using @, and U must appear as a father using *. The chain of $ and # symbols around them should connect U's child to P via a sibling or parent relation that does not break gender roles.

Step-by-Step Solution (Option A):
Step 1: P @ Q means P is the wife of Q. So P is female, Q is P's husband.Step 2: Q $ T means Q is the brother of T, so Q and T are male siblings.Step 3: T # U means T is the daughter of U. This fixes T as female and U as T's parent. However, this conflicts with Q $ T (brother of T) if we interpret brother as gender-specific on Q only. To recover the standard intended exam logic, the step is usually understood as T is the child of U, focusing on parent child links.Step 4: U * W means U is the father of W, confirming U is male and a parent.Step 5: Under the usual reasoning test convention, when Q is the brother of T and T is a child of U, both Q and T are treated as children of U. Thus, Q is also a child of U.Step 6: Since P is the wife of Q and Q is a son of U, U is the father of P's husband, which is exactly father-in-law of P.

Verification / Alternative check of other options:
Consider option B: P @ Q $ T # W * U. Here, U is the father of W, and T is the daughter of W, but there is no requirement that Q is a child of U; instead Q is a sibling of T. This does not force U to be the father of P's spouse. Option C and option D similarly break the needed chain from U through P's spouse: they may make U a parent or a relative of someone else, but not specifically the father of P's husband. Only option A allows U to be interpreted as the father of Q, who is P's spouse.

Why Other Options Are Wrong:
Option B places U as father of W and W as parent of T, so U is a grandparent of T but not clearly linked as parent of P's spouse.Option C makes U a parent of T but connects T to Q in a way that does not make Q the child of U.Option D links U and T, and T and W, but fails to connect U directly as the father of P's spouse.None of these is wrong because option A already correctly expresses the desired relationship.

Common Pitfalls:
Students often decode only the first part P @ Q and then assume that any expression with P married to someone and U appearing as a father must show a father-in-law relation. However, without a clear parent-child link between U and P's spouse, that shortcut fails. The correct method is to follow the entire chain, making sure that the person whose father is U is exactly the person married to P.

Final Answer:
The expression that correctly indicates that U is the father-in-law of P is P @ Q $ T # U * W (option A).