Discrete Mathematics

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

• Symmetric Relation: If , then .
• Antisymmetric Relation: If and , then .

When a relation is both symmetric and antisymmetric, these definitions must hold simultaneously.
Step 1: Apply the symmetric property.
If a pair belongs to relation , then the pair must also belong to .
Step 2: Apply the antisymmetric property.
According to antisymmetry, if both and are present in the relation, then:
Step 3: Combine both conditions.
If both properties hold, the only possible pairs are those where: Thus, the relation can contain only pairs of the form: 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.