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

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

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

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

Prop:

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

b) A~~ = A

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

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