Given S={0,1,2,3,4,5}, find the partition induced by the equivalence relation R where R={(0,0),(0,4),(1,1),(1,3),(4,5),(0,5),(5,4),(5,0),(5,5),(2,2),(3,1),(3,3),(4,0),(4,4)}.
Hey guys, after reading over my book and looking at some examples I came up with..............
We Find the partition of S induced by R or the quotient of S by R.
We recall that If R is the equivalence relation on S and a ∈ S, then the equivalence class [a] is the set of all elements of S to which a is related and [a] = {x = (a, x) ∈ R}
We first find the each of the equivalence classes of 0 [0] = {0,4,5}
The elements related to 1 are [1] = {1,3}
Finally, 2 is only related to itself thus [2] = {2}
S/R = {[0],[1],[2]}
I am not sure if this is incorrect or if it is correct, why its correct. It seems that this is what they want as the book as has similar problem and this is how they answered it. But why does induced equate to the quotient? Why is this answer important?
