Expert: Paul Klarreich - 6/23/2010

Question
if A-{a,b,c}

then the relation on A X A
R={(a,a),(b,b),(c,c)}

is a equvalance reletion.please explain it

Answer
Questioner: pratap
Category: Advanced Math
Private: No
Subject: reletion
Question: if A-{a,b,c}

then the relation on A X A
R={(a,a),(b,b),(c,c)}

is a equvalance reletion.please explain it
...................................
Start with the definition:

A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive.

for all a, b and c in A:

a ~ a. (Reflexivity)
if a ~ b then b ~ a. (Symmetry)
if a ~ b and b ~ c then a ~ c. (Transitivity)

Simply check these things out on your set R.  In your R, xRx will satisfy all three:

Reflex:  Each x is 'R' itself.

Symm:  You do not have any case where  xRy but not yRx, since you only have xRx's.

Trans:  Does xRy and yRz imply  xRz?  Since you can only write things like:

Does aRa and aRa imply  aRa?  

You can answer that.