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

- $A \$ B$ means $A$ is not greater than $B$
- $A \% B$ means $A$ is not smaller than $B$
- $A @ B$ means $A$ is neither smaller nor greater than $B$
- $A \& B$ means $A$ is neither equal to nor smaller than $B$
- $A \# B$ means $A$ is neither equal to nor greater than $B$

Given the statements:

$V \& Y \% K$; $U @ V \$ W \# X$; $R \% J @ W$

Determine which of the following conclusions is/are definitely true:

I. $V \$ R$

II. $K \# U$

III. $R \& U$

Question

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

- $A \$ B$ means $A$ is not greater than $B$
- $A \% B$ means $A$ is not smaller than $B$
- $A @ B$ means $A$ is neither smaller nor greater than $B$
- $A \& B$ means $A$ is neither equal to nor smaller than $B$
- $A \# B$ means $A$ is neither equal to nor greater than $B$

Given the statements:

$V \& Y \% K$; $U @ V \$ W \# X$; $R \% J @ W$

Determine which of the following conclusions is/are definitely true:

I. $V \$ R$

II. $K \# U$

III. $R \& U$

Accepted Answer

**Solution:** 1. Understand the symbols: - $A \$ B$: $A$ is not greater than $B$ (i.e., $A \leq B$) - $A \% B$: $A$ is not smaller than $B$ (i.e., $A \geq B$) - $A @ B$: $A$ equals $B$ - $A \& B$: $A$ is neither equal to nor smaller than $B$ (i.e., $A > B$) - $A \# B$: $A$ is neither equal to nor greater than $B$ (i.e., $A < B$) 2. Analyze the statements: - $V \& Y \% K$ - $V \& Y$ means $V > Y$ - $Y \% K$ means $Y \geq K$ - From these, $V > Y \geq K$ implies $V > K$ - $U @ V \$ W \# X$ - $U @ V$ means $U = V$ - $V \$ W$ means $V \leq W$ - $W \# X$ means $W < X$ - So, $U = V \leq W < X$ - $R \% J @ W$ - $R \% J$ means $R \geq J$ - $J @ W$ means $J = W$ - So, $R \geq J = W$ 3. Evaluate conclusions: - I. $V \$ R$ means $V \leq R$ - From above, $V > K$ and $R \geq J = W \geq V$ (since $V \leq W$ and $W = J$) - Since $U = V \leq W = J \leq R$, it follows $V \leq R$ - So, conclusion I is true. - II. $K \# U$ means $K < U$ - We have $V > K$ and $U = V$, so $U = V > K$ - Hence, $K < U$ is true. - III. $R \& U$ means $R > U$ - From above, $R \geq J = W \geq V = U$ - $R$ may be equal to or greater than $U$, but not definitely greater (since $R \geq U$ but not necessarily $R > U$) - So, conclusion III is not definitely true. **Answer:** Both conclusions I and II are definitely true.

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