I am asked
The relation X on the set $\{1, 2, 3, 4, 5\}$ is defined by the rule $(a, b) ϵ X$ if 3 divides a – b.
- List the elements of X
These are $\{(4,1),(1,4),(5,2),(2,5),(1,1),(2,2),(3,3),(4,4),(5,5)\}$
- List the equivalence class$\color{red}{\text{es}}$
The answer is $\{1,4\},\{2,5\},\{3\}$
This is where I am confused. I thought equivalence class meant that one should only present the elements that don't result in a similar result.
However, $4-1 = 5-2 = 3$ these are all the same
Would anyone care to explain how the equivalence class can be found in this case?
Best Answer
I believe you are mixing up two slightly different questions. Each individual equivalence class consists of elements which are all equivalent to each other. That is why one equivalence class is $\{1,4\}$ - because $1$ is equivalent to $4$. We can refer to this set as "the equivalence class of $1$" - or if you prefer, "the equivalence class of $4$".
Note that we have been talking about individual classes. We are now going to talk about all possible equivalence classes. You could list the complete sets, $$\{1,4\}\quad\hbox{and}\quad\{2,5\}\quad\hbox{and}\quad\{3\}\ .$$ Alternatively, you could name each of them as we did in the previous paragraph, $$\hbox{(the equivalence class of $1$)}\quad\hbox{and}\quad \hbox{(the equivalence class of $2$)}\quad\hbox{and}\quad \hbox{(the equivalence class of $3$)}\ .$$ Or if you prefer, $$\hbox{(the equivalence class of $4$)}\quad\hbox{and}\quad \hbox{(the equivalence class of $2$)}\quad\hbox{and}\quad \hbox{(the equivalence class of $3$)}\ .$$ You see that the "names" we use here are three elements with no two equivalent. I think you are confusing this with the previous paragraph.
Hope this helps!