## Math 3000 - Lecture 10

### Naive Set Theory

Members, Subsets, Equality, Proper Subsets, Power Sets
**Axiom** : There is an empty set.

**Thm**: The empty set is unique.

**Prop**: The empty set and the set itself are subsets.

**Prop**: Transitivity of inclusion.

**Def**. *Union, Intersection, relative complement*

Universal set

Russell's paradox and the Halting Problem

**Prop**:

a) A = and A = A

b) A B A

c) A A B

d) A B = B A and A B = B A

e) A (B C) = (A B) C and A ( B C) = ( A B) C

f) A A = A = A A

g) If A B then A C B C and A C B C.

**Prop**:

a) (A B)^{~} = A^{~} B^{~} and (A B)^{~} = A^{~} B^{~}

b) A^{~~} = A

c) A - B = A B^{~}

d) A B iff B^{~} A^{~}.