What are the Equivalent Classes?

Question

Let A = {a, b, c, d, e, f, g, h, i} and R be a relation on A as follows:

R={(a,a), (f,c), (b,b), (c,f), (a,d), (c,c), (c,i), (d,a), (b,e), (i,c), (e,b), (d,d), (e,e), (f,f), (g,g), (h,h), (i,i), (h,e), (a,g), (g,a), (d,g), (g,d), (b,h), (h,b), (e,h), (f,i), (i,f)}

I know it is the equivalence relation which is symmetric, transitive and reflexive but I am confused about equivalence classes? What are the equivalence classes? How can I find the equivalence classes of the relation?

Answer 1

As you stated, an equivalence relation is a relation which is symmetric, reflexive, and transitive. The definition for those terms is as follows:

Symmetric:

Given a,b in A, if a = b then b = a.

Reflexive:

Given a in A, a = a.

Transitive:

Given a,b,c in A, if a = b and b = c, then a = c.

Using these definitions, we can see that the R relation set in your question is indeed the equivalence relation on A. This is because for every a,b,c in A:

a = a, which is represented by (a,a) in R

if a = b, then b = a, represented by (b,a) and (a,b) both being in R

if a = b and b = c, then a = c, represented by (a,b), (b,c), and (a,c) in R.

You can check to make sure this is true, but I'm pretty sure it is. This is what makes R the equivalence relation. Once we have a definition for an equivalence relation, we can define an equivalence class as follows:

The set of all elements in a set which are equal under a given equivalence relation. In formal notation, {x in S | x -> a}, where -> is the equivalence relation.