Can you help me understand this. Please look at the diagram below.
According to the book, the distinct equivalence classes are... The three distinct equivalence classes are $\{0, 3\}$, $\{1\}$, and $\{2\}$. These form a partition of $\{0, 1, 2, 3\}$.
Why aren't $\{0\}$ and $\{3\}$ part of the equivalence classes?
Thank you,
Regards, Sanone
$\{0\}$ is not an equivalence class as such because there is an element of the original set which is not in $\{0\}$ but which is equivalent to an element of $\{0\}$.
More symbolically $\exists x,y$ with $x\not\in \{0\}, y\in \{0\}, x\sim y$ which contradicts $x \sim y \iff [x]=[y]$
Similarly $\{3\}$ is not an equivalence class as such, for the same reason