Understanding Symbol Meanings and Inequalities

This question asks us to analyze given statements involving specific symbols that represent inequality relationships. We need to translate these symbols into standard mathematical inequality signs to determine which of the given conclusions definitely follow from the statements.

Decoding the Symbols

Let's first understand the meaning of each symbol as provided:

• ‘P$Q’ means ‘P is smaller than Q’ which translates to P < Q
• ‘P©Q’ means ‘P is greater than Q’ which translates to P > Q
• ‘P#Q’ means ‘P is equal to Q’ which translates to P = Q
• ‘P@Q’ means ‘P is either equal to or greater than Q’ which translates to P ≥ Q
• ‘P*Q’ means ‘P is either equal to or less than Q’ which translates to P ≤ Q

Translating the Statements

We are given the following statements:

Statement 1: D $ E $ F $ G

Using the symbol meanings, this translates to:

D < E and E < F and F < G

Combining these, we get a chain relationship: D < E < F < G

Statement 2: K © F

Using the symbol meanings, this translates to:

K > F

Now we have the combined information:

D < E < F < G and K > F

Analyzing the Conclusions

We need to check if the given conclusions are definitely true based on the combined statements.

Conclusion I: K * G

Using the symbol meanings, K * G means K ≤ G.

From our statements, we know K > F and F < G. We have the relationships K > F and F < G. To determine the relationship between K and G, consider the chain K > F < G. There is a break in the direction of the inequality sign (> followed by <). When the inequality signs are in opposite directions between two elements, we cannot establish a definite relationship between them.

For example:

• If F = 5, K = 10, G = 8. Here K > F and F < G is true, and K > G (10 > 8).
• If F = 5, K = 10, G = 12. Here K > F and F < G is true, and K < G (10 < 12).
• If F = 5, K = 10, G = 10. Here K > F and F < G (as 5 < 10) is true, and K = G (10 = 10).

Since the relationship between K and G can be >, <, or =, the conclusion K ≤ G (K is either less than or equal to G) is not definitely true.

Conclusion II: K © D

Using the symbol meanings, K © D means K > D.

From our statements, we know K > F and F > D (since D < F is equivalent to F > D). We have the chain relationship K > F > D. Since K is greater than F, and F is greater than D, it logically follows that K must be greater than D.

This can be seen as K is "greater than" F, and F is "greater than" D. The "greater than" property is transitive. Therefore, K is "greater than" D.

Thus, the conclusion K > D is definitely true.

Conclusion Summary

Based on the analysis:

• Conclusion I (K ≤ G) is not definitely true.
• Conclusion II (K > D) is definitely true.

Therefore, only Conclusion II is definitely true.