Here is the way it is done.
Suppose that is a partition of the set then the equivalence relation on determined by is .
Does that help?
Lets A = {1,2,3,4,5,6,7,8,9,10} and let P be the partition of A given by
P = {{1,2},{3,4,5},{6,7},{8,10},{9}}
Give the equivalence relation R on A that is induced by P.
I am not sure if I am doing this write. Can someone tell me if this is correct or let me know if I am missing anything? Thanks.
R is reflexive iff For all x ͼ A, xRx.
P={(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8) ,(9,9),(10,10)}
R is symmetric iff For all x,y ͼ A, if xRy then yRx.
P={(1,2),(2,1),(3,4),(4,3),(4,5),(5,4),(5,3),(3,5) ,(6,7),(7,6),(8,10),(10,8)}
R is transitive iff For x,y,z ͼ A, if xRy and yRz then xRz.
(3,4)ͼA, (4,5)ͼA and so (3,5)ͼA.
(5,4)ͼA, (4,3)ͼA and so (5,3)ͼA.
I think this is all of the possible transitive possibilities but I also dunno if I have to list them all, I would assume so.
Define the equivalence relation S by S = {(x,y)ͼAxA | 5 divides (x^2-y^2)}.
Determine the partition of A that is induced by S.
I am not exactly sure how to do this one but from what I gather from my textbook it seems to sorta start like this..
Given
Am I doing this correctly? On the right track?