Equivalence Relations

[ragnes]

1. Let R be a relation on X that satisfies
a) for all a in X, (a,a) is in R
b) for a,b,c in X, if (a,b) and (b,c) in R, then (c,a) in R.
Show that R is an equivalence relation.



2. In order for R to be an equivalence relation, the following must be true:
1) for all a in X, (a,a) is in R
2) for a,b in X, if (a,b) is in R, then (b,a) is also in R
3) for a,b,c in X, if (a,b) and (b,c) is in R, then (a,c) is in R.



3. The first part is given by the definition of an equivalence relation. I'm stuck on proving part b. Help please!

 

[lanedance]

how about considering (a,b) and (b,b) for the second part?

 

[ych22]

how about considering (a,b) and (b,b) for the second part?

Proof by contradiction, using lanedance's example.

Or consider (b,c) and (c,c).