Let R be an equivalence relation on s set A. Prove each of the statements in 36-41 directly from the definitions of equivalence relation and equivalence class without using the results of Lemma 8.3.2, Lemma 8.3.3, or Theorem 8.3.4.

For every a and b in A, if a ∈ [ b ] then [ a ] = [ b ] .

To prove:

For all a and b in A, if a ∈ [ b ] then [ a ] = [ b].

Given information:

Let R be an equivalence relation on a set A.

Proof:

R is an equivalence relation on a set A.

Let a∈A and b∈A such that a∈[b].

First part ⊆.

Let x∈[a].

By the definition of [a]={x∈A|x R a}:

x R a

By the definition of [b]={x∈A|x R b} and a∈[b]:

a R b

Since R is an equivalence relation, R is reflexive, symmetric and transitive.

Using the definition of transitive on x R a and a R b, we then obtain:

x R b

By the definition of [b]={x∈A|x R b}

x∈[b]

Since x∈[a] implies x∈[b], the definition of a subset tells us:

[a]⊆[b]

Second part ⊇