Equivalance Relations- ordered pairs

Question

Let $A = \{a, b, c, d\}$
Find the number of equivalence relations on A having:

(a) exactly 4 ordered pairs.
(b) exactly 5 ordered pairs.
(c) exactly 6 ordered pairs

Answer 1

Case 1) Exactly four ordered pair has only one choice $$\{(a,a),(b,b),(c,c),(d,d)\}$$ due to reflexive property requirement.

Case 2) Exactly five ordered pairs has no choice because of symmetric requirement.

Case 3)Exactly six ordered pairs has $6$ choices. We just add two symmetrical new pairs , say $\{(a,b), (b,a)\}$ to the ordered pairs of case $1$ and there are $6$ such ordered pairs.