Question:
For any two sets A and B, prove that
(i) $B \subset A \cup B$
(ii) $A \cap B \subset A$
(iii) $A \subset B \Rightarrow A \cap B=A$
Solution:
(i) For all x ∈ B
x ∈ A or x ∈ B
x ∈ A ∪ B (Definition of union of sets)
B ⊂ A ∪ B
(ii) For all x ∈ A ∩ B
x ∈ A and x ∈ B (Definition of intersection of sets)
x ∈ A
A ∩ B ⊂ A
(iii) Let A ⊂ B. We need to prove A ∩ B = A.
For all x ∈ A
x ∈ A and x ∈ B (A ⊂ B)
x ∈ A ∩ B
A ⊂ A ∩ B
Also, A ∩ B ⊂ A
Thus, A ⊂ A ∩ B and A ∩ B ⊂ A
A ∩ B = A [Proved in (ii)]
∴ A ⊂ B ⇒ A ∩ B = A