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} }