Thread: 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?

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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

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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 *?

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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.

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Please someone help me out with these problems.

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.