Constructing Proofs: Solving Set Functions with Characteristic Functions

[SpatialVacancy]

Constructing Proofs help!

Here is the problem:

Given a set S and subset A, the characteristic function of A, denoted \chi_A, is the function defined from S to \mathbb{Z} with the property that for all u \ \epsilon \ S:

<br /> \chi_A(u)=<br /> \begin{cases}<br /> 1 &amp; \text{if u $ \epsilon \ A$} \\<br /> 0 &amp; \text{if u $ is not \ \epsilon \ A$}<br /> \end{cases}<br />
​

Show that each of the following holds for all subsets A and B of S and all u \ \epsilon \ S.

a. \chi_{A \cap B}(u)= \chi_A (u) \cdot \chi_B (u)
b. \chi_{A \cup B}(u)= \chi_A (u) + \chi_B (u) - \chi_A (u) \cdot \chi_B (u)



I have NO IDEA what this problem is asking...can someone please help!

 

[Galileo]

The brute straightforward way is:
Compute both sides of the equations for all cases:
(I) u is in A and B
(II) u is in A or B, but not both.
(III) u is not in A and not in B.

 

[xanthym]

.
You might present the proof in "Truth Table" format and show equivalence via Table equality. For example:

.
.-----------------------> [B]Χ(A ∩ B)[/B]

                           Member
                             B
                      YES         NO

             YES       [B]1          0[/B]
     Member   
       A
             NO        [B]0          0[/B]




.----------------------> [B]Χ(A)*Χ(B)[/B]

                           Member
                             B
                      YES         NO

             YES    (1)*(1)    (1)*(0)
     Member            [B]1          0 [/B]
       A
             NO     (0)*(1)    (0)*(0)
                       [B]0          0[/B]