# Question on equivalence relations and equivalence classes

• Jan 13th 2010, 02:03 PM
cooltowns
Question on equivalence relations and equivalence classes
Hi i need help with the following question

Let X={1,2,3,4} and let A be the set $\displaystyle X$ x $\displaystyle X$. Define a relation on A by
(x,y)R(w,z)<=>x+y=w+z
Show that R is an equivalence relation on A and describe (as subsets of A) the equivalence classes of R.

The first bit involving that the statement is a equivalence relation, i manged to do the following however i am not sure if it is correct though.

transitive since (x,y)R(x,y)=x+y=x+y
symmetric since (w,z)R(x,y)=w+z=x+y
transitive since (x,y)R(w,z) and let (w,z)R(u,v) ==> (x,y)R(u,v)=x+y=u+v

However i have no idea on how to do the equivalence class stuff.

thanks
(Happy)
• Jan 13th 2010, 02:49 PM
Plato
Quote:

Originally Posted by cooltowns
Let X={1,2,3,4} and let A be the set $\displaystyle X$ x $\displaystyle X$. Define a relation on A by
(x,y)R(w,z)<=>x+y=w+z
Show that R is an equivalence relation on A and describe (as subsets of A) the equivalence classes of R.
transitive since (x,y)R(x,y)=x+y=x+y reflexive?
symmetric since (w,z)R(x,y)=w+z=x+y
transitive since (x,y)R(w,z) and let (w,z)R(u,v) ==> (x,y)R(u,v)=x+y=u+v
However i have no idea on how to do the equivalence class stuff.

What pair(s) is(are) equivalence to (1,2)?
• Jan 14th 2010, 03:40 AM
cooltowns
yep sorry thats a mistake i was meant to say reflexive for (x,y)R(x,y)=x+y=x+y

as i don't understand what you mean by what pairs are equivalent to (1,2) ?

as in do you just mean that it is (2,1) since both (1,2) and (2,1) have the same elements hence equivalent ?
• Jan 14th 2010, 05:06 AM
emakarov
The question is supposed to mean: what pairs are in relation R with (1,2)? Since R is an equivalence relation, one can say (maybe informally) that pairs related by R are equivalent.
• Jan 14th 2010, 09:30 AM
cooltowns
i am not sure on how to answer.

what so specific about (1,2) what the other numbers 3 and 4 ?

(1,2) i presume this means 1+2=1+2 if symmetric
not sure on about symmetric and transitive relations though since if present it like (1,2)R(3,4) 1+2 does not equal 3+4

i am just guessing here i am not really sure on how to approach the question though.
• Jan 14th 2010, 09:37 AM
Plato
Quote:

Originally Posted by cooltowns
i am not sure on how to answer.

what so specific about (1,2) what the other numbers 3 and 4 ?

(1,2) i presume this means 1+2=1+2 if symmetric
not sure on about symmetric and transitive relations though since if present it like (1,2)R(3,4) 1+2 does not equal 3+4

i am just guessing here i am not really sure on how to approach the question though.

Is it true that $\displaystyle (1,2)\mathcal{R}(2,1)?$
If so, then WHY?

What pairs are related to $\displaystyle (1,3)?$
• Jan 14th 2010, 09:54 AM
cooltowns
(1,2)R(2,1) since 1+2=3 and 2+1=3

(1,3) therefore is it related to (3,1) and (2,2) since when you add them up they all equal ?
• Jan 14th 2010, 10:20 AM
Plato
Quote:

Originally Posted by cooltowns
(1,2)R(2,1) since 1+2=3 and 2+1=3

(1,3) therefore is it related to (3,1) and (2,2) since when you add them up they all equal ?

Correct! Now finish.
• Jan 14th 2010, 11:55 AM
cooltowns
ok i am not sure exactly on how to finish the question

but am i correct in saying all these are the combination of relations ?
(1,1) =2
(1,2),(2,1)=3
(1,3),(3,1),(2,2)=4
(1,4),(4,1),(2,3),(3,2)=5
(4,2),(2,4),(3,3)=6
(4,3),(3,4)=7
(4,4)=8

I know that for something to expressed in an equivalence class the notat [] is used. however i am just not sure on how to apply it to my question.
• Jan 14th 2010, 12:02 PM
Plato
Quote:

Originally Posted by cooltowns
ok i am not sure exactly on how to finish the question

but am i correct in saying all these are the combination of relations ?
(1,1) =2
(1,2),(2,1)=3
(1,3),(3,1),(2,2)=4
(1,4),(4,1),(2,3),(3,2)=5
(4,2),(2,4),(3,3)=6
(4,3),(3,4)=7
(4,4)=8
CORRECT!
I know that for something to expressed in an equivalence class the notat [] is used. however i am just not sure on how to apply it to my question.

As for the notation that depends on your instructor/textbook.
Here are two $\displaystyle [(1,3)]_{\mathcal{R}}=\mathcal{R}_{(1,3)}=\{(1,3),(3,1),( 2,2)\}$
• Jan 14th 2010, 12:28 PM
cooltowns
thanks plato for all your help.