Sets, Subsets, Power Sets, Cardinality

Subset (⊆)

A set S is called a subset of a set T (S ⊆ T) if all elements of S are also elements of T {1, 2, 3} ⊆ {1, 2, 3, 4} N ⊆ Z Every natural number is an integer Z ⊆ R Every integer is a real number

Comparing Cardinalities

By definition, two sets have the same size if there is a way to pair their elements off without leaving any elements uncovered (elements don't have to be the same)

Infinite Set Cardinality

Consider |N| where N is set of all natural numbers then we define ℵ0 = aleph-null, aleph-zero, aleph-nought = |N|

Cantor's Theorem

Every set is strictly smaller than its power set. if S is a set, then |S| < |℘(S)| note: not all infinite sets have the same size, there is no biggest ∞, there are ∞-many ∞'s

infinite sets

N = {0, 1, 2, ...} set of all natural numbers (pos integers) Z = {..., -1, 0, 1, ...} set of al integers R = {e, 4, pi, ...} set of all real numbers

Cardinality

The cardinality of a set is the number of elements it contains |A| = number of elements in set A = cardinality if A = {1, 2, 3}, then |A| = 3

Explain why: Not all elements of a set are subsets of that set, and vice versa

This is saying that if you took any element out of a set then you can't say that this element has to also be a subset of that set. This is mostly because the element you took out is likely not a set, so it can't be a subset. Consider S = {1, 2} 1 ∈ S (1 is an element of S) but 1 ⊄ S (1 is not a subset of S) and for the vice-versa part S = {1, 2} {1} ⊆ {1, 2} since 1 is in both but {1} ∉ {1, 2} since S only contains integers 1, 2, not a set containing the integer 1, S would have to contain {1, {1}} in order for the unique case to be true.

Distinction between ∈ and ⊆

We say S ∈ T if, among the elements of T, *one* of them is exactly the *object S* We say S ⊆ T if *S is a set* and every element of S is also an element of T

Union U

all elements of both sets (intersecting elements show up as only one element in union) if A = {1, 2, 3}, B = {3, 4, 5} then A U B = {1, 2, 3, 4, 5}

Intersection (upside down U - ∩)

elements in both sets A = {1, 2, 3}, B = {3, 4, 5} then A ∩ B = {3}

Difference (-)

objects that belong to A and not B (for A - B) if A = {1,2,3}, B={3,4,5} then A - B = {1, 2}

Symmetric difference (Delta, ∆)

objects that belong to A or B but not to their intersection if A = {1,2,3}, B={3,4,5} then A ∆ B = {1,2,4,5}

Relative complement (/)

same thing as difference objects that belong to A and not B (for A / B) if A = {1,2,3}, B={3,4,5} then A / B = {1, 2}

set membership (∈, ∉)

set S, object X x ∈ S (x is contained in S) x ∉ S (x isn't in S)

Power Set (℘(S))

set of all subsets of S ℘(S) = {T | T ⊆ S} Ex. S = {a, b} ℘(S) = {Ø, {a}, {b}, {a, b}} ^^^ could also be {b, a}

set { ... }

unordered collection of distinct objects, which may be anything -no duplicates -equality is determined by elements not ordering

Set Builder Notation

{x | some property x satisfies} Ex: {n | n ∈ N and n is even} set of all even natural numbers ({0, 2, 4, ...})

empty set (Ø or {})

{} or Ø note: 1 ≠ {1}, Ø ≠ {Ø}

Fill in the blank Ø ____ S (subset statement) Ø ____ S (element statement) Ø ____ {Ø} (equality statement) ℘(Ø)____ {Ø} (equality statement)

Ø ⊆ S (elements of Ø are always in S, since there are none) Ø ∉ S (S must contain Ø, S = {Ø, ...}) Ø ≠ {Ø} (element is not same as set containing element ℘(Ø) = {Ø} (subset is set containing Ø, so {Ø}={Ø})

empty set subset property "Vacuous Truth"

Ø ⊆ S where S is any set, is always true "All elements of Ø (empty set) are also elements of S" Vacuous Truth Statement: "All objects of type P are also of type Q" is vacuously true if there are no objects of type P (empty set) "Every element of Ø is also an element of S" "All unicorns are pink"