Proofs Question: Equivalence Relation and Classes

[rdr3]

Homework Statement

Define a relation ~ on ℝ by

a~b if and only if a-b∈Q.

i) Show that ~ is an equivalence relation.

ii) Show that

[a]+=[a+b]

is a well-defined addition on the set of equivalence classes.

Homework Equations

Q represents the set of rational numbers.
An Equivalence Relation must be Reflexive (a~a), Symmetric (a~b implies b~a), and Transitive (a~b and b~c implies a~c).

The Attempt at a Solution

So this is what I came up with and for the most part I think I'm right. If anyone notices anything wrong or needs more information, please let me know and I'll do what I can.

i) Reflexive: a~a → a-a=0 → 0∈Q

Symmetric: a~b → b~a

Here I just showed how if a-b∈Q, then so is it's negative. Thus leading to b~a.

a~b → a-b∈Q → -(a-b)∈Q → -a+b∈Q → b-a∈Q → b~a

Transitive: a~b and b~c → a~c

Here I showed that a-b∈Q and b-c∈Q added together will give a-c∈Q. Showing that a~c.

a-b∈Q and b-c∈Q

(a-b)+(b-c)∈Q → a-b+b-c∈Q → a-c∈Q → a~c

ii) [a]={c₁ | a~c₁ <-> a-c₁∈Q}
={c₂ | b~c₂ <-> b-c₂∈Q}

[a]+=[a+b]
(a-c₁)+(b-c₂)=(a+b)-(c₁+c₂)∈Q

[Side note: (a+b)∈ ℝ and (c₁+c₂)∈ ℝ ]

 

[lippyka]

Therefore, [a+b]={c1+c2 | a~c1 and b~c2 <-> (a+b)-(c1+c2)∈Q }Thus, [a]+=[a+b] is a well-defined addition on the set of equivalence classes.