Question

If a relation is both symmetric and antisymmetric, what must be true about the elements in that relation?

• A. All elements must be comparable
• B. The relation must be transitive
• C. Only pairs of the form \( (a,a) \) can exist
• D. The relation must contain all ordered pairs

Hint:

If a relation is both {symmetric} and {antisymmetric}, the only possible pairs are \((a,a)\). Any pair like \((a,b)\) with \(a \neq b\) would violate antisymmetry.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks about the nature of a relation \(R\) that satisfies both the symmetric and antisymmetric properties simultaneously.
Step 2: Key Formula or Approach:
We need to use the formal definitions of symmetric and antisymmetric relations.

Symmetric: For all \(a, b\), if \( (a,b) \in R \), then \( (b,a) \in R \).

Antisymmetric: For all \(a, b\), if \( (a,b) \in R \) and \( (b,a) \in R \), then \( a = b \).

Step 3: Detailed Explanation:
Let's assume an arbitrary pair \( (a,b) \) is in the relation \(R\).

Because the relation is symmetric, if \( (a,b) \in R \), it must be that \( (b,a) \in R \).

Now we have both \( (a,b) \in R \) and \( (b,a) \in R \).

Because the relation is also antisymmetric, the presence of both \( (a,b) \) and \( (b,a) \) in \(R\) implies that \( a = b \).

This means that for any pair \( (a,b) \) in the relation, it must be the case that \(a=b\). Therefore, the only pairs that can exist in such a relation are of the form \( (a,a) \).
Step 4: Final Answer:
If a relation is both symmetric and antisymmetric, any pair \( (a,b) \) in the relation must satisfy \(a=b\). Thus, only pairs of the form \( (a,a) \) can exist.