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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) = ∅

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
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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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.
Transcribed Image Text: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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