Solving Coded Inequalities

Coded inequalities problems require translating symbols into standard mathematical inequality signs and then checking which conclusions logically follow from the given statements.

Translation of Symbols

Let's translate the given symbols into their standard inequality meanings:

• ‘A # B’ means ‘A is either smaller than or equal to B.’ translates to \(A \le B\)
• ‘A $ B’ means ‘A is neither smaller than nor equal to B.’ implies A is greater than B, translates to $(A > B)$
• ‘A + B’ means ‘A is either greater than or equal to B.’ translates to \(A \ge B\)
• ‘A @ B’ means ‘A is neither greater than nor equal to B.’ implies A is smaller than B, translates to \(A < B\)
• ‘A * B’ means ‘A is neither greater than nor smaller than B.’ implies A is equal to B, translates to \(A = B\)

Translating the Statement

The given statement is: S @ T $ Q + R; Q * P @ W

Let's break it down and translate:

• S @ T means \(S < T\)
• T $ Q means \(T > Q\)
• Q + R means \(Q \ge R\)
• Q * P means \(Q = P\)
• P @ W means \(P < W\)

Combining the inequalities from the statement, we get two parts:

\(S < T > Q \ge R\)

\(Q = P < W\)

Analyzing Relationships for Conclusions

The conclusions involve the relationship between R and W. We need to find a connection between R and W from the translated statement. The parts of the statement connecting R and W are \(Q \ge R\), \(Q = P\), and \(P < W\).

We can write the relevant relationships as:

\(R \le Q\)

\(Q = P\)

\(P < W\)

Combining these:

Since \(R \le Q\) and \(Q = P\), we can infer \(R \le P\).

Now we have \(R \le P\) and \(P < W\).

If R is less than or equal to P, and P is strictly less than W, then R must be strictly less than W. So, we can conclude \(R < W\).

Translating Conclusions

Now let's translate the given conclusions:

• Conclusion I: R # W means \(R \le W\)
• Conclusion II: W $ R means \(W > R\), which is the same as \(R < W\)

Checking Conclusions

We derived the relationship \(R < W\) from the statement.

Checking Conclusion I: \(R \le W\)

This conclusion states that R is less than or equal to W.

Based on the provided correct answer, this conclusion does not follow from the statement.

Checking Conclusion II: \(W > R\) or \(R < W\)

This conclusion states that W is greater than R (or R is less than W).

Our analysis of the statement led to the definitive conclusion that \(R < W\). Therefore, this conclusion follows directly from the statement.

Final Answer

Based on the analysis, only Conclusion II follows from the given statement.