For any two sets A and B, prove that

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 ∈ A ∩ B

 x ∈ A and x ∈ B              (Definition of intersection of sets)

 x ∈ A

 A ∩ ⊂ A

(iii) Let A ⊂ B. We need to prove A ∩ A.

For all x ∈ A

x ∈ A and x ∈ B          (A ⊂ B)

 x ∈ A ∩ B 

 A ⊂ A ∩ B    
  
Also, A ∩ ⊂ A

Thus, A ⊂ A ∩ B and A ∩ ⊂ A

 A ∩ A         [Proved in (ii)]

∴ A ⊂ ⇒ A ∩ A

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