In the following question, two symbolic statements are given: Statements: (I) a R b (II) b R a Conclusions: I. The relation R is symmetric. II. R is a relation (on some set). Assuming the two statements about a and b are true, which of the given conclusions logically follow?

Introduction / Context:
This is a reasoning question about the properties of a relation R. You are told that a R b and b R a are both true and asked whether you can conclude that R is symmetric and that it is a relation. The question tests your understanding of definitions in discrete mathematics and how much can be inferred from a small amount of information.

Given Data / Assumptions:

• Statement (I): a R b is true for some elements a and b.

• Statement (II): b R a is true for the same pair of elements.

• Conclusion I: The relation R is symmetric.

• Conclusion II: R is a relation (on some underlying set).

• A symmetric relation is usually defined on a set S such that for all x and y in S, if x R y then y R x.

Concept / Approach:
A relation R on a set is simply a set of ordered pairs (x, y) where x and y belong to some underlying set. From the given statements, we know that the ordered pairs (a, b) and (b, a) are in R. We ask: does this limited information prove that R is symmetric for all possible pairs, or only shows that R is at least some relation that happens to contain these two pairs?

Step-by-Step Solution:
Step 1: From a R b and b R a, we can confidently assert that R is a relation that contains at least two ordered pairs: (a, b) and (b, a). So it is meaningful to speak of R as a relation. Step 2: Check Conclusion II: "This is a relation." Since R is already used with ordered pairs, it meets the basic definition of a relation, so Conclusion II follows. Step 3: Now examine Conclusion I: "This relation is symmetric." Symmetry requires that for every x and y in the underlying set, if x R y holds then y R x must also hold. However, we only know about one specific pair (a, b). We do not know what happens for any other pair (x, y) different from (a, b). Step 4: It is possible that R includes (a, b) and (b, a) but fails symmetry for some other elements, for example (c, d) is in R but (d, c) is not. The given information does not exclude such possibilities. Step 5: Therefore, the data is insufficient to conclude that R is symmetric overall. We only know that symmetry holds for this one pair a and b.

Verification / Alternative check:
Construct a counterexample. Let the underlying set be {a, b, c}. Define R = {(a, b), (b, a), (c, a)}. Here a R b and b R a are both true, but c R a is true while a R c is false. So R is not symmetric as a whole. Since this example satisfies the given statements yet violates symmetry, Conclusion I cannot be guaranteed by the given data.

Why Other Options Are Wrong:
Option A claims only symmetry follows, which we have shown is not justified. Option C claims that no conclusion follows, but we clearly know R is a relation because we are given two valid instances of it. Option D claims that both conclusions follow, which again wrongly asserts symmetry from insufficient information.

Common Pitfalls:
A common error is to confuse "R is symmetric for a specific pair a and b" with "R is symmetric as a property for all pairs". One example does not prove a universal property. Always check whether you are being asked to prove a general rule or just to check a specific instance.

Final Answer:
Only the fact that R is a relation follows logically, so only Conclusion II is valid.