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 ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks.
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 ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks.
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 ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks.
Transcribed Image Text:10. Proof: Suppose A, B, and C are any sets.
We will show that (A U B) N CCAU(BN C).
Suppose x is any element (A U B) N C.
By definition of intersection x is in A U Band x is in C.
Then by definition of union x is in A or x is in B, and
in both cases x is in C. It follows by definition of
union that in case x is in A and x is in C, then x is in
AU (B N C) by virtue of being in A. And in case x is
in BN C, then x is in A U (B NC) by virtue of being
in BN C. Thus in both cases x is in A U (BN C),
which proves that every element in (A U B) N C is in
AU (B N C).
Hence (A U B) nCCAU(BNC) by definition of
subset.
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