Set Theory

This is a discussion on Set Theory within the C++ Programming forums, part of the General Programming Boards category; How do I draw two logic circuits to show that a + a' = 1? I know how to do ...

  1. #1
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    Exclamation Set Theory

    How do I draw two logic circuits to show that a + a' = 1?

    I know how to do one
    Code:
    a -|\
       | |-
    a'-|/
    the figure meaning "or". How do I draw a second one?

    ==============================================

    Is this right?

    If an element of B happens to be a = "All giraffes have long necks," what would be the dual of B. Make sure that the properties of being a dual hold.

    All things have short necks except for giraffes

    ==============================================

    As stated before, the symbols +, *, 0, and 1 can represent anything that satisfies the four axioms.

    1. Both + and * are communtative. That is, for all b and a in B, a + b = b + a, and a * b = b * a.
    2. There exist special elements in B, 0 and 1, such that for any a in B, a + 0 = a, and a * 1 = a.
    3. For each element a in B, there is a special element, a' such that a + a' = 1 and a*a' = 0. This special element is called the dual of a.
    4. Each operation distributes over the other; this is, for any a, b, and c in B, a*(b+c) = (a*b)+(a*c) and a + (b*c) = (a+b) * (a+c)

    For instance, we can talk about the union and intersections of sets.

    In set theory, what should the symbols 0 and 1 stand for?

    They can't stand for true and false b/c that is boolean isn't it?
    So would it be + and *?

    ==============================================

    Does the set theory model satisfy all the axioms? In particular, give an argument that explains why Axiom 1 is true for the set theory model.

    I say, it does but I have no idea how to make an argument (short). I mean if they mean + and *, then just they are right? I don't know.

    ==============================================

    Please someone help me out with these problems.

  2. #2
    Confused Magos's Avatar
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    Like the picture I attached? I don't know any other way to print that circuit without using other components (which I doubt the question was).
    And I don't see how this proves anything. To see that A + A' = 1, you should use a table or algebra.
    Code:
    A  A'  A + A'
    ------------
    0  1   1
    1  0   1
    Attached Images Attached Images  
    MagosX.com

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  3. #3
    Blank
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    I think it's possible to let 0 = the empty set and then let
    1 = {1}. From there it's clear that 0 U a = a. 0 ^ a = 0. Commutative and distribute laws are satisified. What you
    want to do is show that this or another construction models the axioms.

  4. #4
    Hamster without a wheel iain's Avatar
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    Logic and discrete Mathematics (A computer science perspective) from prentice hall is a good book for set theory.
    Monday - what a way to spend a seventh of your life

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