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Let S be a subset of a universal set U. The characteristic
Chapter 1, Problem 71E(choose chapter or problem)
Let \(S\) be a subset of a universal set \(U\). The characteristic function \(f_{S}\) of \(S\) is the function from \(U\) to the set \(\{0,1\}\) such that \(f_{S}(x)=1\) if \(x) belongs to \(S\) and \(f_{S}(x)=0\) if \(x\) does not belong to \(S\). Let \(A\) and \(B\) be sets. Show that for all \(x \in U\)
,
a) \(f_{A \cap B}(x)=f_{A}(x) \cdot f_{B}(x)\)
b) \(f_{A \cup B}(x)=f_{A}(x)+f_{B}(x)-f_{A}(x) \cdot f_{B}(x)\)
c) \(f_{\bar{A}}(x)=1-f_{A}(x)\)
d) \(f_{A \oplus B}(x)=f_{A}(x)+f_{B}(x)-2 f_{A}(x) f_{B}(x)$\)
Equation Transcription:
Text Transcription:
S
U
f_S
{0, 1}
f_S(x) = 1
f_S(x) = 0
x
A
B
f_A cap B (x) = f_A(x) f_B(x)
f_A cap B(x) = f_A(x) + f_B(x) − f_A(x) f_B(x)
F_bar A(x) =1-f_A(x)
f_A oplus B(x) = f_A(x) + f_B(x)-2f_A(x)f_B(x)
Questions & Answers
QUESTION:
Let \(S\) be a subset of a universal set \(U\). The characteristic function \(f_{S}\) of \(S\) is the function from \(U\) to the set \(\{0,1\}\) such that \(f_{S}(x)=1\) if \(x) belongs to \(S\) and \(f_{S}(x)=0\) if \(x\) does not belong to \(S\). Let \(A\) and \(B\) be sets. Show that for all \(x \in U\)
,
a) \(f_{A \cap B}(x)=f_{A}(x) \cdot f_{B}(x)\)
b) \(f_{A \cup B}(x)=f_{A}(x)+f_{B}(x)-f_{A}(x) \cdot f_{B}(x)\)
c) \(f_{\bar{A}}(x)=1-f_{A}(x)\)
d) \(f_{A \oplus B}(x)=f_{A}(x)+f_{B}(x)-2 f_{A}(x) f_{B}(x)$\)
Equation Transcription:
Text Transcription:
S
U
f_S
{0, 1}
f_S(x) = 1
f_S(x) = 0
x
A
B
f_A cap B (x) = f_A(x) f_B(x)
f_A cap B(x) = f_A(x) + f_B(x) − f_A(x) f_B(x)
F_bar A(x) =1-f_A(x)
f_A oplus B(x) = f_A(x) + f_B(x)-2f_A(x)f_B(x)
ANSWER:Solution:
Step 1:
The given condition for S be a subset of universal set U.A new function is called 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(x) = 0 if x does not belong to S. Let A and B be sets.