Let X be a set of three elements ${a,b,c}$.
1.How many different relations can you define?
The answer is 9, as $R\subset X \times X$
This is easy to see for me, I imagine it as a ~ a, a ~ b, …, c ~ a, …
2.How many different equivalent relations can you define?
The answer is five. The argument is that you can list all partitions: {{a},{b},{c}},{{a,b},{c}},{{a},{b,c}},{{a,c},{b}},{{a,b,c}}.
What I'm not following is that to have an equivalence relation I must show that it's reflexive (i.e. a ~ a), symmetric (i.e. a ~ b = b ~ a) and transitive (i.e. a ~ b, b ~ c = a ~ c). Which for me covers one equivalence relation; involving three elements. How do the rest four look like?!
Best Answer
Your answer for part (1) is incorrect. Any relation on a set $X$ is a subset of $X \times X$. Thus the number of relations is the size of the power set of $X \times X$. In this particular case, $|X \times X|=9$, thus the power set will have size $2^9=512$ relations.
For part (2): Partitions of a set are in one-one correspondence with the equivalence relations on the set, i.e. for each partition, there is an equivalence relation and vice versa.
For example, if we have the partition $\{\{a,b\}, \{c\}\}$, then the elements living in one "part" of the partition will be equivalent to each other. Thus the relation corresponding to this partition will be $$R=\{(a,a), (a,b), (b,a), (b,b), (c,c)\}.$$
Likewise, the relation for the partition $\{\{a\},\{b\}, \{c\}\}$, the corresponding relation will be $$S=\{(a,a), (b,b), (c,c)\}.$$