Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement: For all sets A, B, and C, (A-C) ∩ (B−C) ∩ (A - B) = ∅ Please do the proof the same way as the example (with definiton and stuff). Thanks. Proof by contradiction: Suppose not. Thatis, suppose there exist sets A and B such that(AN B) N (A N B) #Ø. Then there is an element xin (A N B) N (AN B°). By definition of intersection,xE (A N B) and x E (A N B°). Applying the definitionof intersection again, we have that since x E (AN B),xE A and x E B, and since x E (AN B°), xE A andx ¢ B. Thus, in particular, x E B and x B, which isa contradiction. It follows that the supposition is false,and so (A N B)N (AN Bº) = Ø.

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Answered: Use element argument to prove the…

Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement: For all sets A, B, and C, (A-C) ∩ (B−C) ∩ (A - B) = ∅

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement:

For all sets A, B, and C, (A-C) ∩ (B−C) ∩ (A - B) = ∅

Please do the proof the same way as the example (with definiton and stuff). Thanks.

Proof by contradiction: Suppose not. That
is, suppose there exist sets A and B such that
(AN B) N (A N B) #Ø. Then there is an element x
in (A N B) N (AN B°). By definition of intersection,
xE (A N B) and x E (A N B°). Applying the definition
of intersection again, we have that since x E (AN B),
xE A and x E B, and since x E (AN B°), xE A and
x ¢ B. Thus, in particular, x E B and x B, which is
a contradiction. It follows that the supposition is false,
and so (A N B)N (AN Bº) = Ø.

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