Question

Using properties prove that A Union (A Union B)=A

Answer 1

Let x ∈ A ∩ (A ∪ B) 
or x ∈ A and x ∈ A ∪ B 
or x ∈ A and either x ∈ A or x ∈ B 
or x ∈ A, 
since, if x ∈ A, we automatically have that either x ∈ A or x ∈ A. 

Therefore, an element x is in A ∩ (A ∪ B)​ iff it is in A, so A ∩ (A ∪ B) = A.
Here in both the part we dont neet to show the reverse is also true.