# Equivalence Relations

1. Aug 15, 2011

### gtfitzpatrick

1. The problem statement, all variables and given/known data

Deciede if the following are equivalence relations on Z. If so desribe the eqivalence classes
i) a$\equiv$ b if $\left|a\right|$ = $\left|b\right|$
ii) a$\equiv$ b if b=a-2
2. Relevant equations

3. The attempt at a solution

i) $\left|a\right|$ = $\left|a\right|$ so its reflexive

$\left|a\right|$ = $\left|b\right|$ is equivalent to $\left|b\right|$ = $\left|a\right|$ so its symmetric

$\left|a\right|$ = $\left|b\right|$ and $\left|b\right|$ = $\left|c\right|$ then $\left|a\right|$ = $\left|c\right|$ for all values a,b and c elemets of Z so its transitive.

Are there infinite equivalence classes??

ii) a=a so its reflexive
b=a-2 $\neq$ a=b-2 so its not symetric, am i right in thinking this?
Thanks for reading
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 15, 2011

### vela

Staff Emeritus
Yes. Can you describe them? Simply listing a few to show the pattern would be sufficient.
a=a-2?
Yes.

3. Aug 15, 2011

### SammyS

Staff Emeritus
For (i):
What elements(s) of Z is/are equivalent to 3?
What elements(s) of Z is/are equivalent to 7?
What elements(s) of Z is/are equivalent to 0?
What elements(s) of Z is/are equivalent to -5?
...​

For (ii):
This relation is not transitive either.​

4. Aug 15, 2011

### gtfitzpatrick

Thanks for the replies.
So i need to say there are infinity equivalent classes such as -3 equivalent to 3; -5 equivalent to 5 or 10 is equivalent to -10 under the relation.

for ii) i only need to show 1 of the 3 properties doesnt hold, right? or should i show whether all 3 hold or not just for clarity?

5. Aug 15, 2011

### vela

Staff Emeritus
Basically, yes, though your instructor may cringe at your grammar. ;)

The equivalence classes are subsets consisting of all elements that are equivalent to each other. So in this case, they'd be {0}, {1,-1}, {2,-2}, and so on.
Right. You need to show only one requirement doesn't hold to rule out the relation being an equivalence relation.

6. Aug 15, 2011

### gtfitzpatrick

Grammer isnt a strong point of mine :)
Thanks a mill

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