Step 1: Understand the definition of a reflexive relation.
A relation on a set with elements is reflexive if every element is related to itself, i.e., for all . This means the diagonal pairs must be included in .
Step 2: Determine the total number of possible pairs.
The total number of ordered pairs where is , as there are choices for and choices for . This represents all possible elements in the relation .
Step 3: Account for the reflexive condition.
Since must be reflexive, the pairs for to are fixed and must be included. This leaves the off-diagonal pairs where . The number of such pairs is:
These pairs can either be included in or not, giving 2 choices (included or not included) for each pair.
Step 4: Calculate the number of reflexive relations.
The number of ways to choose which of the off-diagonal pairs are included in is . Since the diagonal pairs are mandatory, the total number of reflexive relations is:
Step 5: Verify the result.
For , set , only is required, and there is 1 reflexive relation, .
For , set , diagonal pairs are fixed, and off-diagonal pairs give relations, which matches.