The number of relations, defined on the set \ a, b, c, d\ , which...

For a set with $n = 4$ elements, reflexive condition requires all diagonal pairs $(a,a), (b,b), (c,c), (d,d)$ to be included.
For symmetric relations, non-diagonal pairs come in symmetric pairs. Number of such pairs = $\binom{4}{2} = 6$.
For each pair, we either include both elements or exclude both → 2 choices each.
Total reflexive and symmetric relations = $2^6 = 64$.