sets

A set is an unordered collection of elements or members.

Definition

a set is donated by {} and is defined by listing elements or describing the elements in mathematical notation, For example:

• finite set: $A=$ {$1,2,3$}, where $A$ is a set with 3 elements.
• infinite set: $B=$ {x\mid$$x is an even integer},represents the set of all even integers. The set can represent any set, such as:
• numbered sets
• geometric objects
• functions
• matrices and vectors
• sequence and series
• etc… Sets can be compared by using Set Equivalence.

Properties

• Uniqueness: each element only appears once in a set.
• Unordered: The order does not matter, so {1,2} $\equiv$ {2,1}.

Set Notation

Empty Set notation:($\varnothing$)

• A set that contains no elements

roster or list notation:

• lists all elements of the set in {}(i.e. $C=\{2,4,6,8,10\}$ represents a set with 5 elements)

set-builder notation:

• describes the properties that define the elements of a set(i.e. $D=$ {x\mid$$x is a prime number})

universal set:($U$)

• includes all elements under consideration, and all other sets are subsets of this universal set.

Set Operations

Union($\cup$)

• the set of all elements in either $A$ or $B$

Intersection($\cap$)

• the set of all elements in common between $A$ and $B$

Deference($\setminus$)

• for $A\setminus B$, the set of elements in $A$ but not in $B$

Subset($\subseteq$)

• for $A\subseteq B$, $A$ is a subset if every element of $A$ is also an element in $B$

Symmetric Deference($\oplus$)

• the set of elements $x$ where, $x$ is in $A$ and not in $B$, or $X$ is in $B$ and not in $A$
  • $A\oplus B=(A\setminus B)\cup (B\setminus A)$

Cardinality($|S|$)

• the number of elements in a set.
  • for finite sets, this is a non-negative integer.
  • for infinite sets, cardinality refers to the size of the infinity (e.g. countable vs. uncountable infinity).

Power set($P(S)$)

• the set of all subsets of S(the empty set counts as a subset)

Compliment

• the set of all elements in the universal set $U$ that are not in A.

Disjoint

• two sets are disjoint if they contain no common elements.

Observations

Power Set Size - the size of a power set if base set cardinality is n