In the following questions, symbols ©, #, %, $, and @ are used with the following meanings:

- $M \ © \ N$ 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 the statements:

$$L \ # \ M \ © \ N \ @ \ O \ % \ P; \quad N \ S \ Q \ @ \ T; \quad G \ # \ H \ © \ T$$

Determine which of the following conclusions are definitely true:

I. $O \ $ \ Q$

II. $P \ % \ G$

III. $H \ © \ L$

Choose the correct option:

Question

In the following questions, symbols ©, #, %, $, and @ are used with the following meanings:

- $M \ © \ N$ 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 the statements:

$$L \ # \ M \ © \ N \ @ \ O \ % \ P; \quad N \ S \ Q \ @ \ T; \quad G \ # \ H \ © \ T$$

Determine which of the following conclusions are definitely true:

I. $O \ $ \ Q$

II. $P \ % \ G$

III. $H \ © \ L$

Choose the correct option:

Accepted Answer

**Solution:** 1. Analyze the given statements: - $L \ # \ M \ © \ N \ @ \ O \ % \ P$ - $L \ # \ M$ means $L \leq M$ - $M \ © \ N$ means $M < N$ - $N \ @ \ O$ means $N = O$ - $O \ % \ P$ means $O > P$ Combining these, we get: $$L \leq M < N = O > P$$ 2. Next, $N \ S \ Q \ @ \ T$ means $Q = T$ (since $S$ is not defined, we assume it is a typo or irrelevant; focus on $Q \ @ \ T$ which means $Q = T$). 3. $G \ # \ H \ © \ T$ means: - $G \leq H$ - $H < T$ 4. Now, check each conclusion: - I. $O \ $ \ Q$ means $O \geq Q$ - Since $O = N$ and $Q = T$, and from above $H < T$ and $G \leq H$, but no direct relation between $O$ and $Q$ is given. - However, since $N = O$ and $Q = T$, and from $H < T$ and $G \leq H$, we can infer $O > P$ and $Q = T$. - No direct contradiction, so $O \geq Q$ can be true. - II. $P \ % \ G$ means $P > G$ - From $O > P$ and $G \leq H < T = Q = O$, so $G < O$. - Since $P < O$ and $G < O$, but no direct comparison between $P$ and $G$, so $P > G$ is not definitely true. - III. $H \ © \ L$ means $H < L$ - From $L \leq M < N = O > P$ and $G \leq H < T$, no direct relation between $H$ and $L$. - So $H < L$ is not definitely true. **Answer:** Only conclusion I is definitely true. Hence, option (e) is correct.

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