In the following question, symbols ©, #, %, $, and @ represent different relations as follows:

- $M \ © \ N$ means $M$ is smaller than $N$
- $M \ # \ N$ means $M$ is either smaller than or equal to $N$
- $M \ % \ N$ means $M$ is greater than $N$
- $M \ $ \ N$ means $M$ is either greater than or equal to $N$
- $M \ @ \ N$ means $M$ is neither smaller than nor greater than $N$ (i.e., $M = N$)

**Given Statements:**

$$B \ $ \ D \ % \ G \ @ \ Q; \quad L \ # \ M \ © \ N \ # \ Q; \quad M \ $ \ P \ % \ U$$

**Conclusions:**

I. $D \ % \ U$

II. $N \ % \ G$

III. $M \ © \ B$

Which of the following conclusions definitely hold true based on the above statements?

Question

In the following question, symbols ©, #, %, $, and @ represent different relations as follows:

- $M \ © \ N$ means $M$ is smaller than $N$
- $M \ # \ N$ means $M$ is either smaller than or equal to $N$
- $M \ % \ N$ means $M$ is greater than $N$
- $M \ $ \ N$ means $M$ is either greater than or equal to $N$
- $M \ @ \ N$ means $M$ is neither smaller than nor greater than $N$ (i.e., $M = N$)

**Given Statements:**

$$B \ $ \ D \ % \ G \ @ \ Q; \quad L \ # \ M \ © \ N \ # \ Q; \quad M \ $ \ P \ % \ U$$

**Conclusions:**

I. $D \ % \ U$

II. $N \ % \ G$

III. $M \ © \ B$

Which of the following conclusions definitely hold true based on the above statements?

Accepted Answer

**Solution:** 1. Understand the meaning of each symbol: - $M \ © \ N$: $M < N$ - $M \ # \ N$: $M \leq N$ - $M \ % \ N$: $M > N$ - $M \ $ \ N$: $M \geq N$ - $M \ @ \ N$: $M = N$ 2. Analyze the given statements: - $B \ $ \ D$: $B \geq D$ - $D \ % \ G$: $D > G$ - $G \ @ \ Q$: $G = Q$ From these, we get: $$B \geq D > G = Q$$ - $L \ # \ M$: $L \leq M$ - $M \ © \ N$: $M < N$ - $N \ # \ Q$: $N \leq Q$ Since $Q = G$, and $N \leq Q$, we have: $$L \leq M < N \leq Q = G$$ - $M \ $ \ P$: $M \geq P$ - $P \ % \ U$: $P > U$ So: $$M \geq P > U$$ 3. Evaluate conclusions: - **Conclusion I: $D \ % \ U$ (Is $D > U$?)** From above: $$B \geq D > G = Q$$ and $$M \geq P > U$$ Since $D > G = Q$ and $U < P \leq M$, but no direct relation between $D$ and $U$ is given. However, since $D > G = Q$ and $N \leq Q$, and $M < N$, and $M \geq P > U$, it implies $D$ is greater than $U$ because $D > G = Q \geq N > M \geq P > U$. So, **Conclusion I is true**. - **Conclusion II: $N \ % \ G$ (Is $N > G$?)** From above, $N \leq Q$ and $G = Q$, so $N \leq G$. Therefore, $N > G$ is false. - **Conclusion III: $M \ © \ B$ (Is $M < B$?)** From above: $$L \leq M < N \leq Q = G < B \geq D$$ Since $B \geq D > G = Q \geq N > M$, it follows that $M < B$. Hence, **Conclusion III is true**. **Answer:** Both conclusions I and III are definitely true.

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