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  • If A and B are subsets of the universal set U, then show that (i) A ⊂ A ∪ B (ii) A ⊂ B ⇔ A ∪ B = B (iii) (A ∩ B) ⊂ A

If A and B are subsets of the universal set U, then show that
(i) A ⊂ A ∪ B
(ii) A ⊂ B ⇔ A ∪ B = B
(iii) (A ∩ B) ⊂ A
 

Answers (1)

(i) Given that A \subset U and B \subset U

     To prove: A \subset A\cup B we have to show that if x\in A; x\in A\cup B

    Let x\in A\Rightarrow x\in A \ or x\in B \Rightarrow x\in A\cup B

    hence, A \subset A\cup B

(ii) A\subset B then let x\in A\cup B

    \Rightarrow x \in A \ or \ x\in B

    \Rightarrow x \in B

    A\cup B \subset B \qquad\qquad (i)

    But B \subset A\cup B \qquad\qquad (ii)

    From (i)  and (ii) we get A\cup B = B

    Now if A\cup B = B, then

    let y\in A \Rightarrowy\in A\cup B \Rightarrow y\in B

    Hence, A \subset B

    Thus, A \subset B \Rightarrow A\cup B = B

iii)

\begin{array}{l} \text { Let } x \in A \cap B \\ \quad x \in A \text { and } x \in B \\ \quad x \in A \\ \text { Hence, } A \cap B \subset A \end{array}

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