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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

0 2 Views | Posted 4 months ago
Asked by Shiksha User

  • 1 Answer

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    Answered by

    Vishal Baghel | Contributor-Level 10

    4 months ago

    ( i ) G i v e n     t h a t : A ⊂ U     a n d     B ⊂ U           L e t     x ∈ A     o r     x ∈ B ⇒                 x ∈ A ∪ B         H e n c e ,                 A ⊂ ( A ∪ B ) ( i i ) I f     A ⊂ B               T h e n     l e t     x ∈ A ∪ B ⇒ x ∈ A     o r     x ∈ B ⇒           A ∪ B ⊂ B                                                                     … ( i ) B u t     B ⊂ A ∪ B                                                                     … ( i i ) F r o m     e q . ( i )     a n d     ( i i ) ,     w e     g e t                         A ∪ B = B L e t     y ∈ A           ⇒ y ∈ ( A ∪ B )           ⇒ y ∈ B ⇒           y ∈ B           ⇔ A ∪ B = B ( i i i ) L e t     x ∈ A ∩ B ⇒ x ∈ A     a n d     x ∈ B         ⇒ x ∈ A           S o ,     A ∩ B ⊂ A

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