53. If A is subset of B then find Aintersection B AunionB A-B ​

Question

53. If A is subset of B then find Aintersection
B AunionB A-B

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Arianna 1 month 2021-08-16T14:43:02+00:00 1 Answer 0 views 0

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    2021-08-16T14:44:41+00:00

    Answer:

    Step-by-step explanation:

    The trivial case first: If  ⊆  and  =∅ , then  ∩=∅= . Conversely, if  ∩=  and  =∅ , then  ⊆ , since  ∅  is a subset of every set.

    Now, suppose  ≠∅ . Suppose  ⊆ . If  ∈ , then  ∈ , from the definition of subset. Thus,  ∈  and  ∈ . Therefore, by the definition of intersection of sets,  ∈∩ . Therefore, every    in    is in  ∩ . From the definition of containment,  ⊆∩ . Each element of  ∩  is in both    and   , so  ∩⊆ . Since  ⊆∩  and  ∩⊆ , the definition of set equality gives  =∩ . For the converse. If  ∩= , then since  ∩⊆ ,  ⊆ . Q.E.D.

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