1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Constructing Proofs help

  1. Mar 9, 2005 #1
    Constructing Proofs help!!

    Here is the problem:

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


    [tex]
    \chi_A(u)=
    \begin{cases}
    1 & \text{if u $ \epsilon \ A$} \\
    0 & \text{if u $ is not \ \epsilon \ A$}
    \end{cases}
    [/tex]

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

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



    I have NO IDEA what this problem is asking.....can someone please help!!!!
     
  2. jcsd
  3. Mar 10, 2005 #2

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  4. Mar 10, 2005 #3

    xanthym

    User Avatar
    Science Advisor

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

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

                               Member
                                 B
                          YES         NO

                 YES       [B][COLOR=Red]1 [/COLOR]         0[/B]
         Member  
           A
                 NO        [B]0          0[/B]




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

                               Member
                                 B
                          YES         NO

                 YES    (1)*(1)    (1)*(0)
         Member            [B][COLOR=Red]1[/COLOR]          0 [/B]
           A
                 NO     (0)*(1)    (0)*(0)
                           [B]0          0[/B]
     
     
    Last edited: Mar 10, 2005
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Constructing Proofs help
  1. Help with proofs (Replies: 4)

  2. Help with proofs (Replies: 2)

  3. Help with this proof (Replies: 1)

  4. Constructing a proof (Replies: 3)

Loading...