Understanding the Coding Language

The problem presents a special coding language where symbols represent different family relationships. To solve this, we first need to clearly understand what each symbol signifies:

• 'A $ B' means 'A is the mother of B'
• 'A # B' means 'A is the father of B'
• 'A ! B' means 'A is the sister of B'
• 'A @ B' means 'A is the brother of B'
• 'A * B' means 'A is the wife of B'
• 'A > B' means 'A is the husband of B'
• 'A % B' means 'A is the father-in-law of B'
• 'A ^ B' means 'A is the mother-in-law of B'

Deciphering the Family Relationships from Coded Statements

We are given three coded statements (a, b, and c). We will analyze each one to build a clear picture of the family connections and determine the specific relationship between U and A.

Statement (a): Analyzing 'P > Q $ R > S ! T'

Let's break down this complex statement piece by piece:

• P > Q translates to 'P is the husband of Q'. This also means Q is the wife of P.
• Q $ R translates to 'Q is the mother of R'.
• R > S translates to 'R is the husband of S'. This means S is the wife of R.
• S ! T translates to 'S is the sister of T'.

From these translations, we can make the following deductions:

• Since Q is the mother of R, and P is her husband, P is the father of R.
• Since R is the husband of S, it means S is R's wife.
• The statement 'S ! T' tells us that S and T are siblings.

Statement (b): Analyzing 'U # T'

This statement provides a direct relationship:

• U # T translates to 'U is the father of T'.

Considering that S and T are siblings (from statement a), if U is the father of T, then U must also be the father of S.

Statement (c): Analyzing 'R # A'

This statement establishes a direct link between R and A:

• R # A translates to 'R is the father of A'.

Determining the Relationship: U to A

Now, let's synthesize the information gathered from all three statements to find the relationship of U to A:

1. We know that R is the father of A (established from statement c).
2. We also know that R is married to S (from statement a, 'R > S').
3. Because R is A's father and S is R's wife, S must be the mother of A.
4. From statement b ('U # T') and the fact that S and T are siblings (from statement a), we deduced that U is the father of S.
5. Therefore, since U is the father of S, and S is the mother of A, U is the mother's father of A.

In conclusion, based on the given coding language and statements, U is the mother's father of A.