We first show that A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C)
Let x ∈ A ∪ (B ∩ C). Then
x ∈ A or x ∈ B ∩ C
⇒ x ∈ A or (x ∈ B and x ∈ C)
⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C)
⇒ (x ∈ A ∪ B) and (x ∈ A ∪ C)
⇒ x ∈ (A ∪ B) ∩ (A ∪ C)
Thus,
A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C) ....(1)
Now we will show that
(A ∪ B) ∩ (A ∪ C) ⊂ (A ∪ C)
Let x ∈ (A ∪ B) ∩ (A ∪ C)
⇒ x ∈ A ∪ B and x ∈ A ∪ C
⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C)
⇒ x ∈ A or (x ∈ B and x ∈ C)
⇒ x ∈ A or (x ∈ B ∩ C)
⇒ x ∈ A ∪ (B ∩ C)
Thus,
(A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C) ... (2)
So, from (1) and (2), we have
A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)