Let S be a subset of a universal set U. The characteristic

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)