I'm trying to understand a problem that my textbook gives me. Here is the problem:
The relation $R$ is an equivalence relation on the set $A$. Find the distinct equivalence classes of $R$.
$A = \{0, 1, 2, 3, 4\}$
$R = \{(0,0), (0,4), (1,1), (1,3), (2,2), (3,1), (3,3), (4,0), (4,4)\}$
Here is the solution:
$[0] = \{x \in A | x\;R\;0 \} = \{0, 4\}$
$[1] = \{x \in A | x\;R\;1 \} = \{1, 3\}$
$[2] = \{x \in A | x\;R\;2 \} = \{2\}$
$[3] = \{x \in A | x\;R\;3 \} = \{1, 3\}$
$[4] = \{x \in A | x\;R\;4 \} = \{0, 4\}$
Note that $[0]=[4]$ and $[1]=[3]$. Thus the distinct equivalence classes of the relation are $\{0,4\}$, $\{1,3\}$, and $\{2\}$.
My problem here is that I am not understanding the solution. I do not understand how it came up with an answer for each equivalence class of every element $A$. As in, how is $\{0,4\}$ equal to $\{x \in A | x\;R\;0 \}$, and how is that equal to $[0]$? I can understand that $[0]=[4]$ since they both equal $\{0,4\}$ but I'm not sure how to arrive at that answer.
I'm trying this problem:
$A = \{a, b, c, d\}$
$R = \{(a,a), (b,b), (b,d), (c,c), (d,b), (d,d)\}$
However, I am lost because I do not understand how to arrive at answers for every element in $A$.
Any help explaining this to me would be much appreciated!
Best Answer
for the first problem $0 \sim 4 ,1 \sim 3, 2 \sim 2$ so you have 3 equivalence classes (note that R is an equivalence realation) . for the second one $a \sim a , b \sim d , c\sim c$.