Math 3000 - Lecture 10

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) [emptyset] [intersect] A = and A = A

b) A [intersect] B [is a subset of] A

c) A [is a subset of] A B

d) A B = B A and A [intersect] B = B A

e) A (B C) = (A B) C and A [intersect] ( B C) = ( A B) C

f) A A = A = A [intersect] A

g) If A [is a subset of] B then A C B C and A [intersect] C B C.

Prop:

a) (A B)^~ = A^~ [intersect] B^~ and (A B)^~ = A^~ B^~

b) A^~~ = A

c) A - B = A [intersect] B^~

d) A [is a subset of] B iff B^~ A^~.