Analyzing Logical Inequality Statements

This question asks us to evaluate the truthfulness of two statements based on given relationships between different elements represented by letters. These relationships are expressed using inequality symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), and = (equal to).

Evaluating Statement I

Statement I says: If A ≤ B > C < D > E > F ≥ G = H; then B is always greater than E.

Let's analyze the relationship chain provided for Statement I:

A ≤ B > C < D > E > F ≥ G = H

We are interested in the relationship between B and E. Let's look at the path connecting B and E in the chain:

B > C < D > E

To establish a definite relationship between two elements in an inequality chain, there must be a continuous path between them where the inequality symbols are all facing the same general direction (e.g., all > or ≥, or all < or ≤). In the path from B to E (B > C < D > E), we see a change in the direction of the inequality signs (from > to < between C and D, and from < to > between C and D). For example, B > C and C < D. This means we cannot determine a fixed relationship between B and D. Similarly, since we can't relate B and D definitively, we cannot guarantee a definite relationship between B and E just by looking at this chain.

Let's consider some examples to see if B is *always* greater than E:

• Example 1: Let B = 10, C = 8, D = 12, E = 7. Here B > C (10 > 8), C < D (8 < 12), D > E (12 > 7). In this case, B > E (10 > 7). This fits the condition B > E.
• Example 2: Let B = 10, C = 8, D = 9, E = 11. Here B > C (10 > 8), C < D (8 < 9), D > E (9 > 11) - Wait, 9 > 11 is false. Let's adjust Example 2.
• Example 2 (Corrected): Let B = 10, C = 8, D = 15, E = 12. Here B > C (10 > 8), C < D (8 < 15), D > E (15 > 12). In this case, B > E (10 > 12) - Wait, 10 > 12 is false. Let's adjust Example 2 again to satisfy B > C < D > E but not B > E.
• Example 2 (Corrected again): Let B = 10, C = 8, D = 12, E = 11. Here B > C (10 > 8), C < D (8 < 12), D > E (12 > 11). All parts of the chain B > C < D > E are satisfied. However, in this case, B is not greater than E (10 is not > 11).

Since we found a scenario where the given chain is true but B is not greater than E, Statement I is not always correct.

Evaluating Statement II

Statement II says: If P > Q = R ≥ S = T ≤ U = V > W; then S is always less than V.

Let's analyze the relationship chain provided for Statement II:

P > Q = R ≥ S = T ≤ U = V > W

We are interested in the relationship between S and V. Let's look at the path connecting S and V in the chain:

S = T ≤ U = V

Let's combine these relationships step-by-step:

• S = T: This means S and T are equal. We can replace T with S in the next part.
• T ≤ U: Since S = T, this becomes S ≤ U. S is less than or equal to U.
• U = V: Since U = V, we can replace U with V in the previous relationship.
• Combining S ≤ U and U = V, we get S ≤ V.

The relationship derived between S and V is S ≤ V. This means S can be less than V (S < V) or S can be equal to V (S = V).

Statement II claims that S is *always* less than V (S < V). However, our analysis shows that S can also be equal to V (S = V). If S = V, then S is not strictly less than V.

Let's consider an example where S = V is possible:

• Example: Let S = 5, T = 5, U = 5, V = 5. Here S = T (5 = 5), T ≤ U (5 ≤ 5), U = V (5 = 5). All parts of the chain S = T ≤ U = V are satisfied. In this case, S is equal to V (5 = 5), not strictly less than V.

Since we found a scenario where the given chain is true but S is not strictly less than V (because S can be equal to V), Statement II is not always correct.

Conclusion on Statements

Based on the analysis of both statements:

• Statement I: "B is always greater than E" is not always true.
• Statement II: "S is always less than V" is not always true.

Therefore, neither Statement I nor Statement II is correct.

Revision Table: Checking Inequality Relations

Understanding how to check relationships in inequality chains is key.

Additional Information: Understanding Inequality Logic

In logical reasoning questions involving inequalities, the transitivity property is crucial. Transitivity means if A > B and B > C, then A > C. Similarly, if A < B and B < C, then A < C. This property holds when the inequality signs between elements in a chain are consistently in the same direction (>, ≥) or (<, ≤). However, if there is a change in the direction of the inequality signs (e.g., > followed by <, or < followed by >), transitivity breaks down, and a definite relationship between the elements at the ends of that segment cannot be established without more information.

The presence of an '=' sign between two elements means they are equivalent, and you can essentially treat them as the same element for evaluating relationships with others in the chain (e.g., if A < B = C, then A < C).

To confirm if a statement (e.g., A > E) is *always* true based on a complex chain, you must be able to derive that specific relationship (A > E) or a stronger one (like A ≥ E, provided A can never be equal to E) from the given chain using transitivity and equality rules. If the chain results in a possibility where the statement is false (e.g., A = E or A < E), then the statement is not always true.