Suppose that U is the universal set, and that A, B and C are three arbitrary sets of elements of U. Prove that if C ∖ A = B, then the intersection of A and B is empty. Hint: use an indirect proof.

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Suppose that       U   is the universal set, and that   A,   B and   C are three arbitrary sets of elements of   U. Prove that if   C  ∖  A  =  B, then the intersection of   A and   B is empty. Hint: use an indirect proof.

Taniyah Lin · Accepted Answer

Show   A  =  B implies   A  ∩  B  =  ∅.Assume   A  ∩  B  ≠  ∅  ..Then there is a   x s.t   x  ∈  A and   x  ∈  B.Since   x  ∈  B, and   C , we have   x  ∈  C,then   x  ∉  A, a contradiction.

Filloltarninsv9p · Accepted Answer

The following is a direct proof:Assume   C  −  A  =  BThen, by substitution,  A  ∩  B  =  A  ∩            (        C  −  A            )      By definition of set difference  =  A  ∩            (        C  ∩      A    c              )      By commutative law,  =  A  ∩            (            A    c    ∩  C            )      By associative law,  =            (        A  ∩      A    c              )        ∩  CBy negation law,  =  ∅  ∩  CBy domination law,  =  ∅Therefore, if   C  −  A  =  B, then   A  ∩  B  =  ∅

Suppose that U is the universal set, and that A, B and C are three arbitrary sets of elements of U. Prove that if C∖A=B, then the intersection of A and B is empty. Hint: use an indirect proof.

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