Thread: Still need help with relation problem

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.

Originally Posted by CPR

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.

You've got your set for R incorrect. R will be a set of pairs of ordered pairs. Ie. it will take the form


So, for example (2, 1) R (1, 2) implies that the ordered pair [(2, 1), (1, 2)] belongs to the set R. etc.

-Dan

Originally Posted by CPR

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.

OK. Show us what you have done for yourself.

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.

That is not R, that is AxA.
R relates pairs in AxA to themselves.
(1,4)R(2,2) because (1)(4)=(2)(2).
Now begin again.

I would if I knew what to do. I see what you have written and suggested, but I dont know how to do what you have suggested. how is it that (1,4) R
(2,2) or how does (1)(4) = (2)(2). Are you saying 1x4 = 2x2? How did you get these ordered pairs?

R relates pairs to pairs.
Can you tell which of the are true?
(2,2)R(1,4) T F
(2,3)R(2,4) T F
(2,2)R(2,2) T F
(2,3)R(3,2) T F
(1,3)R(1,4) T F

Originally Posted by CPR

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.

Originally Posted by topsquark

You've got your set for R incorrect. R will be a set of pairs of ordered pairs. Ie. it will take the form


So, for example (2, 1) R (1, 2) implies that the ordered pair [(2, 1), (1, 2)] belongs to the set R. etc.

-Dan

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

T
F
T
T
F

I am know the first one bc you told me. How is (2,2)R (1,4)

Originally Posted by CPR

I am know the first one bc you told me. How is (2,2)R (1,4)

Originally Posted by CPR

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.

s = 2, t = 2
u = 1, v = 4

Since 2*2 = 1*4 we know that (2, 2) R (1, 4). Hence (2, 2) and (1, 4) are in the same equivalence class.

-Dan