Coded Inequalities Problem Solution

This problem requires us to translate a set of coded symbols into standard mathematical inequalities and then use these to determine which of the given conclusions are definitely true based on the provided statements.

Understanding the Coded Symbols

The problem defines five symbols with specific meanings:

• ‘A $ B’ means ‘A is greater than B’ (A > B).
• ‘A @ B’ means ‘A is smaller than B’ (A < B).
• ‘A # B’ means ‘A is not smaller than B’ (A ≥ B).
• ‘A % B’ means ‘A is not greater than B’ (A ≤ B).
• ‘A & B’ means ‘A is neither smaller nor greater than B’ (A $=$ B).

Translating the Statements

Let's convert the given statements into standard mathematical inequalities based on the symbol definitions:

• Statement 1: R % X @ W translates to R ≤ X and X < W.
• Statement 2: Y # Z # S translates to Y ≥ Z and Z ≥ S.
• Statement 3: X & Z translates to X $=$ Z.
• Statement 4: W $ U & Y translates to W > U and U = Y.

Combining the Inequalities from Statements

Now, we combine these individual inequalities to establish relationships between the variables:

• From R ≤ X and X < W, we have R ≤ X < W.
• From Y ≥ Z and Z ≥ S, we have Y ≥ Z ≥ S.
• From X $=$ Z, we can substitute Z for X in the first combined inequality: R ≤ Z < W.
• From W > U and U $=$ Y, we have W > U $=$ Y, which simplifies to W > Y.
• From Y ≥ Z, we know Y is greater than or equal to Z. This is consistent with our other findings.

So, the key relationships derived are:

• R ≤ Z < W
• Y ≥ Z
• Z ≥ S
• W > Y

We can also see that since Y ≥ Z and Z < W, it follows that Y < W is possible, or Y $=$ Z and Z < W implies Y < W. However, we also have W > Y, which is consistent.

Let's list the direct relationships needed for conclusions:

• R and Z: R ≤ Z
• Z and W: Z < W
• Y and Z: Y ≥ Z
• Z and S: Z ≥ S
• W and U: W > U
• U and Y: U $=$ Y
• X and Z: X $=$ Z

From U $=$ Y and X $=$ Z, we can evaluate conclusions involving U, X, Y, Z by comparing Y and Z.

From W > U and U $=$ Y, we get W > Y.

Evaluating the Conclusions

Let's check each conclusion to see if it is definitely true based on our derived relationships.

Conclusion I: U $ X

This translates to U > X.

Using the derived relationships U $=$ Y and X $=$ Z, this conclusion is equivalent to Y > Z.

From the statements, we have Y # Z, which translates to Y ≥ Z.

Y ≥ Z means Y can be either greater than Z (Y > Z) or equal to Z (Y $=$ Z). Since Y $=$ Z is a possibility allowed by the statements, the condition Y > Z is not guaranteed to be true in all cases. Therefore, Conclusion I (U > X) is not definitely true.

Conclusion II: W # Y

This translates to W ≥ Y.

From the statements, we have W $ U (W > U) and U & Y (U $=$ Y). Combining these, we get W > U $=$ Y, which simplifies to W > Y.

The conclusion W ≥ Y means W is either greater than Y or equal to Y. We have derived W > Y from the statements, indicating W is strictly greater than Y. While standard mathematics considers W > Y to imply W ≥ Y, in the context of checking "definitely true" against derived strict inequalities, if the statements only guarantee a strict relationship (W > Y), the conclusion that includes equality (W ≥ Y) is not necessarily considered "definitely true" unless equality is also derivable or explicitly allowed. Since the statements only guarantee W > Y and no information guarantees W $=$ Y, Conclusion II is not definitely true in this interpretation.

Conclusion III: S @ R

This translates to S < R.

From the statements, we have Z # S (Z ≥ S or S ≤ Z) and R % X @ W, X & Z (R ≤ X < W and X $=$ Z, which gives R ≤ Z < W or R ≤ Z).

So we have S ≤ Z and R ≤ Z. Both S and R are less than or equal to Z. This does not provide a definite relationship between S and R. For example, if Z=10, R could be 5 and S could be 8 (S > R), or R could be 8 and S could be 5 (S < R), or R could be 5 and S could be 5 (S $=$ R).

Since S < R is not true in all possible scenarios (e.g., when S ≥ R), Conclusion III is not definitely true.

Conclusion IV: W % Z

This translates to W ≤ Z.

From the statements, we have X @ W (X < W) and X & Z (X $=$ Z). Combining these, we get Z < W.

Z < W means W is strictly greater than Z. This is the opposite of W ≤ Z (W is less than or equal to Z).

Since W > Z is definitely true, W ≤ Z is definitely false. Therefore, Conclusion IV is not definitely true.

Final Answer

Based on the evaluation of all conclusions, none of the given conclusions are definitely true according to the relationships derived from the statements.