# Equivalence Relation and Classes

• May 4th 2010, 05:53 AM
iwish123
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?
• May 4th 2010, 07:17 AM
Plato
Quote:

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 \$\displaystyle n^2\ge 0\$ that is proves the relation is reflexive.

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

There are only three equivalence classes. One is the set \$\displaystyle \{0\}\$.
Now what are the other two?
• May 4th 2010, 07:39 AM
iwish123
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?
• May 4th 2010, 07:48 AM
Plato
Quote:

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

Correct!

Quote:

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.
• May 4th 2010, 08:07 AM
iwish123
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?