Symbols are defined as follows: "A $ B" means "A is the father of B", "A # B" means "A is the daughter of B", and "A @ B" means "A is the sister of B". Which of the following expressions means that O is the father of two children?

Introduction / Context:

This question uses symbolic notation to represent family relationships. You must decode each symbol correctly and then see which expression says that O is the father of two different children. Symbolic relation questions like this are common in reasoning exams and test both your understanding of family relationships and your ability to manipulate simple symbolic languages.

Given Data / Assumptions:

• The symbols are defined as:
  • "A $ B" means "A is the father of B".
  • "A # B" means "A is the daughter of B".
  • "A @ B" means "A is the sister of B".
• We need an expression that shows O is the father of two different children.
• The options are:
  • O @ M # N
  • O # N # M
  • O @ N # M
  • N # O $ M
• We assume standard family logic: siblings share at least one parent, and "father of two children" means both children are explicitly or implicitly shown as children of O.

Concept / Approach:

The main idea is to decode each option using the definitions and see in which expression O appears as a father twice—once for each child. Because the symbols combine in a chain, you must be careful not to reverse the roles of parent and child in each pair. The correct pattern will show O as the parent (on the left side of $) of two family members.

Step-by-Step Solution:

Step 1: Decode option (a) O @ M # N. O @ M → O is the sister of M (O is female, sibling of M). M # N → M is the daughter of N. There is no "$" with O as the parent, so this does not show O as a father of anyone. Step 2: Decode option (b) O # N # M. O # N → O is the daughter of N. N # M → N is the daughter of M. Here O and N are daughters; there is no place where O is a father, so this cannot express "O is father of two children". Step 3: Decode option (c) O @ N # M. O @ N → O is the sister of N. N # M → N is the daughter of M. Again, O is only a sister, not a father, so this option is also incorrect. Step 4: Decode option (d) N # O $ M. N # O → N is the daughter of O. So O is a parent (and specifically the father, according to the $ definition) of N. O $ M → O is the father of M. Combining these, O is the father of both N and M, which means O has two children, N and M.

Verification / Alternative check:

In option (d), the chain N # O $ M creates a small family tree: O is father of M (from O $ M) and also father of N (since N # O means N is daughter of O). Thus O clearly has two children: N (daughter) and M (child, gender unspecified in the symbol). No other option has O on the left-hand side of the $ symbol at all, let alone in a way that produces two children. Therefore, (d) is the only expression that matches the requirement.

Why Other Options Are Wrong:

In options (a), (b) and (c), O is never shown as a father. Instead, O appears as a sister or daughter, which does not satisfy the condition that O must be the father of two children.

Specifically:

• In (a), O is a sister and M is a daughter of N.
• In (b), O and N are daughters in a chain.
• In (c), O is a sister of N, and N is a daughter of M.

None of these describe O as a father figure.

Common Pitfalls:

Many candidates accidentally reverse the meaning of the symbols (for example, reading "A $ B" as "B is the father of A") or fail to note which side of the symbol is the parent. This leads to completely incorrect interpretations. A good practice is always to re-write each symbolic pair in plain English before trying to combine them. Also, make sure you double-check which person must play the role of father—in this case, O—and only then scan the options for the one that presents O as a parent for two different individuals.

Final Answer:

The expression that shows O is the father of two children is N # O $ M.