1. ## 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.

2. 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```

3. 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. Logic and discrete Mathematics (A computer science perspective) from prentice hall is a good book for set theory.