Let $M$ be the set $\{1,2,3\}$. How many Equivalence relations $R \subset M \times M$ exists?
My idea is to count the disjoint partitions of M:
$K_1= \{\{1\},\{2\},\{3\}\}\Leftrightarrow\{(1,1),(2,2),(3,3)\}$
$K_2= \{\{1,2\},\{3\}\} \Leftrightarrow\{(1,1),(1,2),(2,1),(2,2),(3,3)\} $
$K_3= \{\{1,3\},\{2\}\}\Leftrightarrow\{(1,1),(1,3),(3,1),(2,2),(3,3)\}$
$K_4= \{\{1\},\{2,3\}\}\Leftrightarrow\{(1,1),(2,3),(3,2),(2,2),(3,3)\}$
$K_5=\{1,2,3\}\Leftrightarrow K_5=M^2$
So the answer would be $5$. Is this correct? Reflexivity and Symmetry are obvious, but how can i check the right side for Transitivity?