Find the equivalence classes of the relation R = {(0, 0),(1, 1),(1, 2),(2, 2),(2, 1),(3, 3),(3, 4),(4, 3),(4, 4)}
on the set A = {0, 1, 2, 3, 4}.
How do i solve this question. I'm attempting to teach myself at the moment so any help will be appreciated.
As far as i'm aware the equivalence class of a is the set of all elements x in A, such that x is related to a by r.
On
The equivalence class of $x$, denoted $[x]$, is the set of all elements of $A$ that are related to $x$. More formally, $[x] = \{y \in A | (x,y) \in R\}$.
Looking at $R$, we see that $1$ is related to $2$ and $3$ is related to $4$, so we can ‘combine’ the equivalence classes for $1$ and $2$, and for $3$ and $4$. We can ignore all of the other pairs as they are simply the result of the fact that $R$ is an equivalence relation—they don’t give us any more information. We have $[1] = [2]$ and $[3] = [4]$ and so our equivalence classes are
$$[0] = \{0\}$$ $$[1] = \{1,2\}$$ $$[2] = \{1,2\}$$ $$[3] = \{3,4\}$$ $$[4] = \{3,4\}$$
An equivalence relation on $A$ induces a partition on $A$, so we may also show the equivalence classes by writing
$$\{\{0\},\{1,2\},\{3,4\}\}$$
As you said in the question, we form equivalence classes by finding elements in $R$ related to different elements of $A$.
So from the relation $R$ we find that $(0,0) \in R \Rightarrow 0 \in [0].$ Also, $ (1,2) \in R \Rightarrow 2 \in [1]\ \text{and}\ 1 \in [2]$ etc. Therefore, $[0] = \{0\}, [1] = \{1,2\}$ etc.
Here $[\cdot]$ denotes an equivalence class and '$\cdot$' is the representative of the class.
Can you find other classes like $[2]$ and $[3]$ from here?