Symbols Define Relations

• A*B: A is the father of B (A is male).
• A $\Delta$ B: A is the husband of B (A is male, B is female).
• A $\nabla$ B: A is the wife of B (A is female, B is male).
• A $\square$ B: A is the sister of B (A is female).

Target Relation Decoded

The goal is to represent "C is the father-in-law of the sister of D".

• Let the "sister of D" be represented by a person, say F. According to the definition, F $\square$ D, which implies F is female.
• C is the "father-in-law" of F. This means C is the father of F's spouse. Let F's spouse be E.
• So, we need C to be the father of E (C * E), and E to be the spouse of F.
• If E is the husband of F, then E $\Delta$ F. The combined chain is C * E $\Delta$ F $\square$ D.
• If E is the wife of F, then E $\nabla$ F. The combined chain is C * E $\nabla$ F $\square$ D.

Option 4 Solves Puzzle

Let's examine the expression: C * E $\Delta$ F $\square$ D

1. F $\square$ D: F is the sister of D. This correctly establishes F as female.
2. E $\Delta$ F: E is the husband of F. This implies E is male and F is female. This is consistent with F being the sister of D.
3. C * E: C is the father of E. This implies C is male.

Result: C is the father of E, and E is the husband of F. Therefore, C is the father of F's husband. This makes C the father-in-law of F. Since F is the sister of D, this expression accurately represents the required relationship.

Incorrect Options Rejected

Option 2 (C * E $\nabla$ F $\square$ D):

• F $\square$ D implies F is female.
• E $\nabla$ F implies E is the wife of F (E=Female, F=Male).
• This leads to a contradiction: F must be male (as the husband of E) and female (as the sister of D) at the same time. Hence, Option 2 is invalid.

Other Options: Options 1 and 3 result in C being the direct parent (father or mother) of a person in the chain, not the father-in-law of the sister of D.