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

- $M \ © \ N$ means $M$ is smaller than $N$
- $M \ # \ N$ means $M$ is smaller than or equal to $N$
- $M \ % \ N$ means $M$ is greater than $N$
- $M \ $ \ N$ means $M$ is 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:

$$22 \ $ \ 26 \ $ \ 35 \ @ \ 41 \ # \ 29; \quad 53 \ $ \ 29 \ @ \ 32 \ # \ 38$$

Determine which of the following conclusions are definitely true:

I. $35 \ © \ 32$

II. $53 \ $ \ 41$

III. $41 \ $ \ 29$

Question

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

- $M \ © \ N$ means $M$ is smaller than $N$
- $M \ # \ N$ means $M$ is smaller than or equal to $N$
- $M \ % \ N$ means $M$ is greater than $N$
- $M \ $ \ N$ means $M$ is 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:

$$22 \ $ \ 26 \ $ \ 35 \ @ \ 41 \ # \ 29; \quad 53 \ $ \ 29 \ @ \ 32 \ # \ 38$$

Determine which of the following conclusions are definitely true:

I. $35 \ © \ 32$

II. $53 \ $ \ 41$

III. $41 \ $ \ 29$

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:
   - $22 \ $ \ 26$: $22 \geq 26$ (False, but given as true, so accept as per question)
   - $26 \ $ \ 35$: $26 \geq 35$ (False, but given)
   - $35 \ @ \ 41$: $35 = 41$ (False, but given)
   - $41 \ # \ 29$: $41 \leq 29$ (False, but given)
   - $53 \ $ \ 29$: $53 \geq 29$ (True)
   - $29 \ @ \ 32$: $29 = 32$ (False)
   - $32 \ # \ 38$: $32 \leq 38$ (True)

Since the statements are assumed true, we accept them as given.

3. Evaluate conclusions:
   - I. $35 \ © \ 32$ means $35 < 32$ which is false.
   - II. $53 \ $ \ 41$ means $53 \geq 41$ which is true.
   - III. $41 \ $ \ 29$ means $41 \geq 29$ which is true.

4. From the above, conclusion II is definitely true.

5. Conclusion III is true based on the symbol meaning, but the statement $41 \ # \ 29$ (i.e., $41 \leq 29$) was given, which conflicts. However, since the question asks which conclusions are definitely true assuming the statements are true, conclusion III is not definitely true.

6. Conclusion I is false.

7. Therefore, only conclusion II is definitely true, and either conclusion I or III could be true depending on interpretation.

**Answer:** Only II and either I or III is true

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