1.1.1 A is a subset of B if all elements of A are elements of B.
1.1.2 ⊆
1.2 Proper Subset
1.2.1 A is a proper subset of B if A is a subset of B and A is not equal to B
1.2.2 ⊂
2 UNION
2.1 ∪
2.2 A + B
3 INTERSECTION
3.1 ∩
3.2 everything the same in A + B
4 DISJOINT
4.1 A ∩ B = ∅
5 LESS
5.1 A - B (A\B) = A - (A ∩ B)
6 COMPLEMENT
6.1 A complement is everything in U outside of A (U\A)
7 Commutative
8 Associative
9 Distributive
10 De Morgan Laws
10.1 (A ∩ B) complement
= A complement ∪ B
complement
10.2 (A ∪ B)^c = A^c ∩ B^c
11 Involutivity of the Complement
11.1 (A^c)^c) = A
11.2 "An involution is a map such that
applying it twice gives the
(original) identity. Familiar
examples: reflecting across the
x-axis, the y-axis, or the origin in
the plane.
14.1.3 ∅ and {∅} are different objects. ∅ has no elements, whereas {∅} has one element.
14.1.4 P(A) and A are viewed as living in separate world to avoid phenomena like Russell's paradox.
14.1.5 If A has n elements, then P(A) has 2^n elements.
14.1.6 In the ZFC (Zermelo Fraenkel set theory) standard
system, it is an axion of set theory that every set has
a power set, which implies no set consisting of all
possible sets could exist, else what would it power
set be?