# Thread: Equivalence Relation and Classes

1. ## Equivalence Relation and Classes

Hello, I have the following problem, I think I've proved the equivalence relaltion correctly, but cannot figure out how to work the equivalence classes. Hopefully some one will know!

The relation T is defined on the set of integers by:

aTb if and only if ab>0 or a=b=0

For the equivalence realtion I have come up with the following

1. Reflexive: aTa as a=a=0 or we could have stated that aa>0

2. Symmetric: Take integers a,b. aTb means ab>0 or a=b=0 => b=a=0 or ba>0. Hence we also have bTa

3. Transitive: Take integers a,b,c. If we have aTb and bTc, ab>0 or a=b=0 and bc>0 or b=c=0. So we have a=b=c=0, hence, bTc

Then, the next question I was asked was what are the equivalence relations?

2. Originally Posted by iwish123
The relation T is defined on the set of integers by:
aTb if and only if ab>0 or a=b=0

For the equivalence realtion I have come up with the following
1. Reflexive: aTa as a=a=0 or we could have stated that aa>0
2. Symmetric: Take integers a,b. aTb means ab>0 or a=b=0 => b=a=0 or ba>0. Hence we also have bTa
3. Transitive: Take integers a,b,c. If we have aTb and bTc, ab>0 or a=b=0 and bc>0 or b=c=0. So we have a=b=c=0, hence, bTc
Then, the next question I was asked was what are the equivalence relations?
Note that $n^2\ge 0$ that is proves the relation is reflexive.

The suppose that $aTb~\&~bTc$ you want to show $aTc$.
Start with if $b=0$ then at once that means that $a=0=c$ done.
On the other hand, if $b\not= 0$ then $a,~b,~\&~c$ all have the same sign. So $ac>0$.

There are only three equivalence classes. One is the set $\{0\}$.
Now what are the other two?

3. Thank you for the help, would the other 2 equivalence classes be numbers greater than 0 and numbers less than 0? In what format would I write that in exam if im correct?

4. Originally Posted by iwish123
the other 2 equivalence classes be numbers greater than 0 and numbers less than 0?
Correct!

Originally Posted by iwish123
In what format would I write that in exam if im correct?
I have no idea. That strictly depends on your textbook/notes/instructor.
There are many notations for equivalence classes in use.

5. Thanks again.
I've practiced another equivalence relation question involving complex numbers.

The relation R is defined for complex numbers z=x+yi and w=a+bi. zRw if and only if x+b=y+a

I am asked to give the elements of the equivalence class containg [i]

I have wrote, [i]={xeC:xRi}={xeC:x-i}, so would the elements of the equivalence class [i] be those numbers congruent to i or -i, so the elements would be 1 and -1?