I have a question on Equivalent Classes: $A=\{1,2,3,4,5......,20\}$ , $R$ is define on $A$ as follows:
For All $x,y$ element of $A$, $xRy$ if and only if $4|(x-y)$
I have the answer $\{1,5,9,13,17\},\{2,6,10,14,18\},\{3,7,11,15,19\},\{4,8,12,16,20\}$
However the book didn't explain how it derive the answer, Can someone please explain to me. Thank you!
There are $4$ different possible remainders after dividing by $4$: $0,1,2,$ and $3$.
Since these are the only possible remainders, every number has to be in the same equivalence class of one of them. So simply use the definition to check which class each number belongs in.
$$4|(4-0)\quad 4|(5-1)\quad 4|(6-2)\quad 4| (7-3)$$ and so on. This lets us classify every integer into one of four equivalence classes, the one containing $0$, the one containing $1$, the one containing $2$, and the one containing $3$. At some point you’ll probably notice the pattern that lets you shortcut having to check each one individually: the equivalence class of $k$ is the set of all integers that are $k$ more than a multiple of $4$. Notice that this is true even for $k$ not in $\{0,1,2,3\}$!
Now, your question isn’t interested in all integers, only those in $A$. So we throw out all the negative numbers, $0$, and everything bigger than $20$. What’s left is the four sets given by the book.