Sets

https://slides.com/georgelee/ics141-sets/live

We're done with logic!

Sets, Functions, Sequences, Sums, Matrices

Intro to Sets

Group "objects" together

  • Positive integers
  • Rational numbers less than 100
  • Students in this class
  • {1, -2, "Peyton", "Raynor"}
  • {1, 3, {2, 4}}

Properties of Sets

  • Elements are unique (no duplicates)
  • Elements are unordered
  • Can be finite or infinite
  • Can contain other sets

Fancy Notations

  • We write "x ∈ A" if x is a member of set A and "x ∉ S" if x is not a member of set A.
  • We use ∅ to denote the empty set (note that {∅} is a set of one element)
  • For finite sets, the cardinality of a set S is the number of elements in the set and is denoted by |S|.

Tabular Notation

  • Enumerate the values in the set
  • {2,3,5,7}
  • {b, c, a, m, z}
  • Do not do something like {3, 6, 9, 12, ...}
  • Use this form only if you can list all the values

Set Builder Notation

  • { x | p(x) } where if p(x) is true, then x is a member of the set
  • { x ∈ S | p(x) } where the first part describes our universe of discourse (natural numbers, integers, etc.)
  • { x | x ∈ ℕ | x is even and less than 100 }
  • { x | x is a game on Steam }
  • Use this form to describe large or infinite sets

Special Sets

  • ℕ: the set of natural numbers (usually including 0)
  • ℤ: the set of integers
  • ℚ: the set of rational numbers
  • ℝ: the set of real numbers
  • ℂ: the set of complex numbers
  • ℤ+: the set of all positive integers
  • ℤn: the set of all nonnegative integers less than n
  • ℝ+: the set of all positive real numbers

Relationships Between Sets

Subset / Superset

  • We say X is a subset of Y if and only if (iff) all elements of x are also in Y.
  • ∀ x [ ( x ∈ X ) → ( x ∈ Y ) ] 
  • Denoted as X ⊆ Y
  • We say Y is a superset of X iff X is a subset of Y.
  • The empty set ∅ is a subset of all sets.
  • Example: {1, 2, 3} is a subset of {x | x ∈ ℤ }

Equal / Not Equal

  • We say X and Y are equal (X = Y) if X is a subset of Y and Y is a subset of X
  • ∀ x [ ( x ∈ X ) ↔ ( x ∈ Y ) ]
  • If X and Y are not equal, we write X ≠ Y
  • Remember that order does not matter!
    {2, 4, 6} = {6, 4, 2}
  • Be careful, {{2, 4, 6}} is not equal to {2, 4, 6}

Other Relationships

  • X is a proper subset of Y iff X ⊆ Y and X ≠ Y
  • X and Y are said to be (mutually) disjoint iff no element is in both X and Y
  • Example: {1, 3, 5} is a disjoint set of {2, 4, 6}

Operations With Sets

Universal Set

The set of all elements for which the sets we're operating on will be selecting from.

 

For example if U = {x | x ∈ ℤ }, then sets A and B inside of this universe can only contain integers.

Union

The Union of two sets A and B are all of the elements of A and all of the elements in B.

 

Using set builder notation: A ∪ B = { x | x ∈ A ∨ x ∈ B }

 

{2, 4, 6} ∪ {1, 3, 5} = {1, 2, 3, 4, 5, 6}

Intersection

The Intersection of two sets A and B are all of the elements common to both A and B.

 

Using set builder notation: A ∩ B = { x | x ∈ A ∧ x ∈ B }

 

{2, 4, 6} ∩ {1, 2, 3} = {2}

Complement

The Complement of a set A is all of the elements in U that are not in A.

 

Using set builder notation: A = { x ∈ U | x ∉ A ) }

Set Difference

The Difference of two sets A and B are all the elements in A that are not in B. Note that A - B is not usually equal to B - A.

 

Using set builder notation: A − B = { x | x ∈ A ∧ x ∉ B } 

 

{2, 4, 6} - {1, 2, 3} = {4, 6}

Cartesian Product

The Cartesian Product of two sets A and B is a set of ordered pairs containing all pairs of elements between A and B.

 

An ordered pair is of the form (x, y)

 

Using set builder notation: A × B = { (x, y) | x ∈ A ∧ y ∈ B }  

 

{2, 4} x {1, 2} = {(2, 1), (2, 2), (4, 1), (4, 2)}

Power Set

The Power Set of a set A is the set of all subsets of A (including the empty set and itself). May be denoted as ℘(A) or 2^A

 

{9,10,11} = { X | X ⊆ {9,10,11} } = { ∅, {9}, {10}, {11}, {9,10}, {10,11}, {9,11}, {9,10,11} }

 

Sets

By George Lee

Sets

Is the set of all sets a member of itself?

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