Here is my problem:
Let H= {1,2,3,4}, A = HxH, and define a relation on A by (s,t) R(u,v) if and only if st = uv.
1. Show R is an equivalence relation on A.
2. Compute the partition A/R that corresponds to the equivalent relation.
This is what I have:
R= (1,1), (1,2), (1,3),(1,4), (2,1), (2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1), (4,2), (4,3),(4,4).
It is reflexive, symmetric, and transitive. Therefore it is an equivalence relation.
R(1) = {1,2}, R(3) {3,4}
However, I am not sure about the partition A/R. Please help.
This is what I have:
R= (1,1), (1,2), (1,3),(1,4), (2,1), (2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1), (4,2), (4,3),(4,4).
It is reflexive, symmetric, and transitive. Therefore it is an equivalence relation.
However, I do no know exactly how to do the partition A/R. Please help.
This what I think
A/R= {(1,2), (3,4)}
Please explain to me what to do! Ive been struggling with this for 2 days. Im getting really frustrated.
You asked for some extra help, but there isn't really that much more to give.
Showing that R is an equivalence relation on A is easy. To compute the partition, simply find the equivalence classes:
(1, 1)
(1, 2) ~ (2, 1)
(1, 3) ~ (3, 1)
(2, 2) ~ (1, 4) ~ (4, 1)
etc.
-Dan