Proof A U (A ∩ B) ⊆ A: Understanding x∈A

[dirtybiscuit]

Homework Statement

I am trying to prove the absorption law
A U (A ∩ B) = A
I know that a way to prove this is to show that each is a subset of the other but I'm a little confused about one part in the process (below)

Homework Equations

The Attempt at a Solution

Let x∈A U (A ∩ B)
then x∈A or x∈(A ∩ B)
so because x∈A then we know that A U (A ∩ B) ⊆ A (I don't understand why this line is true.)

Why just because x∈A does it mean that A U (A ∩ B) ⊆ A is true? Any help is greatly appreciated.

 

[haruspex]

then x∈A or x∈(A ∩ B)
so because x∈A then we know that A U (A ∩ B) ⊆ A

Arguably there's a step missing in there.
If x∈(A ∩ B) then x∈A , so either way x∈A U (A ∩ B) implies x∈A.
Thus you have shown that every element of A U (A ∩ B) is an element of A. Hence A U (A ∩ B) ⊆ A.

 

[Maged Saeed]

so because x∈A then we know that A U (A ∩ B) ⊆ A (I don't understand why this line is true.)

This is true because each element in the subset ' A U (A ∩ B) ' must belong to A .

 

[RUber]

This is a logical argument. You are trying to show that if x is in $A \cup ( A \cap B)$, then it is also in A, and if x is in A, then it is in $A \cup ( A \cap B)$.
You have already shown the first part (edit) by the definition of the intersection: if x is in $A \cup ( A \cap B)$, then it is also in A, which implies that $A \cup ( A \cap B)\subseteq A$,
Next, you need to show that $A \subseteq A \cup ( A \cap B)$. That should be simple enough by the definition of a union. So it looks like you are just about done.