Coding Logic Explained

The problem presents a specific coding language used to define family relationships:

• 'A + B' signifies 'A is the brother of B'.
• 'A - B' signifies 'A is the mother of B'.
• 'A × B' signifies 'A is the father of B'.
• 'A ÷ B' signifies 'A is the son of B'.

Our task is to decipher the relationships defined by the expression P×Q×R + D÷S - B and identify the statement that does NOT hold true.

Expression Decoding Step-by-Step

Let's analyze the given expression piece by piece:

1. P × Q: According to the rules, 'A × B' means 'A is the father of B'. So, P is the father of Q. We represent this as: P \rightarrow Q.
2. Q × R: Similarly, 'Q × R' means 'Q is the father of R'. Represented as: Q \rightarrow R.
   • Combined Analysis: From P being the father of Q (P \rightarrow Q) and Q being the father of R (Q \rightarrow R), we deduce that P is the paternal grandfather of R.
3. R + D: The rule 'A + B' means 'A is the brother of B'. Therefore, R is the brother of D. This indicates R is male. We represent this sibling relationship as: R \leftrightarrow D.
4. D ÷ S: The rule 'A ÷ B' means 'A is the son of B'. So, D is the son of S. This means D is male. We represent this as: S \rightarrow D.
   • Combined Analysis: Since R and D are siblings (R \leftrightarrow D), and D is the son of S (S \rightarrow D), it follows that S is a parent of both R and D. We know Q is R's father (from step 2). Since S is also a parent of R, and Q is the father, S must be the mother of R and D. So, we have S \rightarrow R and S \rightarrow D.
5. S - B: The rule 'A - B' means 'A is the mother of B'. Thus, S is the mother of B. Represented as: S \rightarrow B.
   • Overall Family Structure: We now know S is the mother of R, D, and B. Since Q is the father of R, and S is the mother of R, Q and S are the parents of R. Consequently, Q and S are also the parents of D and B.
     • P is the father of Q (P \rightarrow Q).
     • Q is the father of R, D, and B (Q \rightarrow R, Q \rightarrow D, Q \rightarrow B).
     • S is the mother of R, D, and B (S \rightarrow R, S \rightarrow D, S \rightarrow B).
     • P is the paternal grandfather of R, D, and B.
     • R is the brother of D and B.

Statements Analysis

Let's examine each provided statement against the deduced family relationships:

Statement 1: R is B's brother.

We concluded that R and B share the same parents (Q and S) and that R is male. This confirms that R is B's brother. This statement is correct.

Statement 2: Q is B's father.

Our analysis in step 5 clearly established that Q is the father of B (along with R and D). This statement is correct.

Statement 3: P is B's mother's father.

B's mother is S. B's father is Q. P is the father of Q. Therefore, P is the father of B's father. The statement claims P is the father of B's mother (S), which contradicts our findings. This statement is NOT correct.

Statement 4: R is P's son's son.

P's son is Q. R is the son of Q. Therefore, R is indeed the son of P's son. This statement is correct.

Identifying Incorrect Statement

After carefully decoding the expression and verifying each statement, we find that the statement "P is B's mother's father" is the only one that is not factually correct based on the given coding language and expression.