Set Operations

SUBSET
1. Basic Subset
  1. A is a subset of B if all elements of A are elements of B.
  2. ⊆
2. Proper Subset
  1. A is a proper subset of B if A is a subset of B and A is not equal to B
  2. ⊂
UNION
1. ∪
2. A + B
INTERSECTION
1. ∩
2. everything the same in A + B
DISJOINT
1. A ∩ B = ∅
LESS
1. A - B (A\B) = A - (A ∩ B)
COMPLEMENT
1. A complement is everything in U outside of A (U\A)
Commutative
Associative
Distributive
De Morgan Laws
1. (A ∩ B) complement = A complement ∪ B complement
2. (A ∪ B)^c = A^c ∩ B^c
Involutivity of the Complement
1. (A^c)^c) = A
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.
Transitivity of Inclusion
1. A ⊆ B ∧ B ⊆ C → A ⊆ C
Equality of Sets
1. from (P ↔ Q) ↔ [(P → Q) ∧ (Q → P)]
2. A = B ↔ (A ⊆ B) ∧ (B ⊆ A)
3. Non-Equality of Sets
  1. A != B ↔ [(A\B) ∪ (B\A)] != 0
POWER SET
1. P(A)
  1. Recall ∅ ⊆ A. Also, A ⊆ A.
  2. If A = {0, 1},
    1. then P(A) = {∅, {0}, {1}, {0,1}}
    2. If A = {a, b, c},
      1. then P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.
      2. If A = ∅,
        P(A) = {∅}
        P(P(A)) = {∅, {∅}}
  3. ∅ and {∅} are different objects. ∅ has no elements, whereas {∅} has one element.
  4. P(A) and A are viewed as living in separate world to avoid phenomena like Russell's paradox.
  5. If A has n elements, then P(A) has 2^n elements.
  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?

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