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

Concept: In discrete mathematics, relations can have different properties. Two important properties are:

• Symmetric Relation: If \( (a,b) \in R \), then \( (b,a) \in R \).
• Antisymmetric Relation: If \( (a,b) \in R \) and \( (b,a) \in R \), then \( a = b \).

When a relation is both symmetric and antisymmetric, these definitions must hold simultaneously.
Step 1: Apply the symmetric property.
If a pair \( (a,b) \) belongs to relation \(R\), then the pair \( (b,a) \) must also belong to \(R\). \[ (a,b) \in R \Rightarrow (b,a) \in R \]
Step 2: Apply the antisymmetric property.
According to antisymmetry, if both \( (a,b) \) and \( (b,a) \) are present in the relation, then: \[ a = b \]
Step 3: Combine both conditions.
If both properties hold, the only possible pairs are those where: \[ a = b \] Thus, the relation can contain only pairs of the form: \[ (a,a) \] These are called reflexive pairs.
Step 4: Conclusion.
Therefore, if a relation is both symmetric and antisymmetric, the relation can contain only ordered pairs where both elements are the same. \[ (a,a) \]