Problem 5. Let S be a subset of a universal set U. The characteristic function fs of S is the function from U to the set {0, 1} such that fs(x) = 1 if x belongs to S and fs = 0 if x does not belong to S. Let A and B be sets. Show that for all x € U, Note: We can prove all of these identities by showing that the left-hand side is equal to the right-hand side for all possible values of x. In each instance, there are four cases to consider, depending on whether x is in A and/or B. 5(a) fanB(x) = fA(x) · fB(x) 5(b) fAUB(x) = fA(x)+ fB(x) – fA(x) · fB(x) %3D 5(c) fA©B(x) = fA(x)+ fB(x) – 2fA(x)fB(x)
Problem 5. Let S be a subset of a universal set U. The characteristic function fs of S is the function from U to the set {0, 1} such that fs(x) = 1 if x belongs to S and fs = 0 if x does not belong to S. Let A and B be sets. Show that for all x € U, Note: We can prove all of these identities by showing that the left-hand side is equal to the right-hand side for all possible values of x. In each instance, there are four cases to consider, depending on whether x is in A and/or B. 5(a) fanB(x) = fA(x) · fB(x) 5(b) fAUB(x) = fA(x)+ fB(x) – fA(x) · fB(x) %3D 5(c) fA©B(x) = fA(x)+ fB(x) – 2fA(x)fB(x)
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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Transcribed Image Text:Problem 5. Let S be a subset of a universal set U. The characteristic function fs of S is the function from
U to the set {0, 1} such that fs(x) = 1 if x belongs to S and fs = 0 if x does not belong to S. Let A and
B be sets. Show that for all x E U,
Note: We can prove all of these identities by showing that the left-hand side is equal to the right-hand
side for all possible values of x. In each instance, there are four cases to consider, depending on whether x
is in A and/or B.
5(a) fanB(x) = fA(x) · fB(x)
5(b) fAUB (x) = fA(x)+ fB(x) – fA(x)· fB(x)
5(c) fAÐB(x) = fA(x)+ fB(x) – 2fA(x)fB(x)
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