Understanding Symbolic Relationships

This question asks us to interpret given symbols that represent mathematical inequalities and equalities and then determine which conclusions are definitely true based on the provided statements.

Decoding the Symbols

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

• $ means 'is smaller than' (<)
• © means 'is greater than' (>)
• # means 'is equal to' (=)
• @ means 'is either equal to or greater than' (≥)
• * means 'is either equal to or less than' (≤)

Analyzing the Statements

We are given the following statements:

E # F $ G $ H

G @ I

Let's translate these statements into standard inequality and equality notation:

Statement 1: E = F < G < H

Statement 2: G ≥ I

Combining the information, we have:

• E = F
• F < G
• G < H
• G ≥ I (which is the same as I ≤ G)

From E = F and F < G, we can deduce E < G.

So, the core relationships are: E < G < H and I ≤ G.

Evaluating the Conclusions

Now let's evaluate each conclusion based on the derived relationships.

Conclusion I: H © I

Translating this conclusion: H > I.

From the statements, we know G < H (from E = F < G < H).

We also know G ≥ I (or I ≤ G).

Let's consider the relationship between H and I using G as a link:

We have I ≤ G and G < H.

Since G is strictly less than H (G < H) and I is less than or equal to G (I ≤ G), it follows that I must be strictly less than H.

If I = G, then I < H.

If I < G, then I < G < H, which means I < H.

In both possible cases from I ≤ G, I is always less than H. Therefore, H > I is definitely true.

Conclusion II: E @ I

Translating this conclusion: E ≥ I.

From the statements, we know E = F and F < G, which means E < G.

We also know G ≥ I (or I ≤ G).

Let's consider the relationship between E and I using G as a link:

We have E < G and I ≤ G.

If G = I, then E < G means E < I. In this case, E ≥ I is false.

If G > I, then E < G. E could be less than I, equal to I, or greater than I. For example, if G=10, I=8, then E < 10. E could be 5 (E < I), or E could be 8 (E = I), or E could be 12 (E > I - though E < G must hold, so E cannot be 12 if G=10). Let's take valid examples: If E=5, G=10, I=8, E < G and G > I, but E < I. This counterexample shows E ≥ I is not always true.

Since there are scenarios where E < I is true based on the given statements, the conclusion E ≥ I is not definitely true.

Summary of Conclusions

• Conclusion I (H > I) is definitely true.
• Conclusion II (E ≥ I) is not definitely true.

Based on our analysis, only Conclusion I is definitely true.