Fundamentals of Sets in Statistics

SETS

Definition and Notation

In mathematics and statistics, a set is a well-defined collection of distinct objects, called elements. Sets are typically denoted by capital letters, and their elements are listed within curly braces.

• Finite and Infinite Sets: Sets may contain a finite or infinite number of elements.

• Example: $A = \{C, A, N, D\}$ is the set of letters in the word 'CAND'.

• Notation: $S = \{a, b, c, ...\}$

Special Sets

• Empty Set: The set with no elements is called the empty set, denoted by $\varnothing$ or $\{\}$.

• Example: $\varnothing \subseteq S$ for any set $S$.

• Set of Real Numbers: $\mathbb{R} = \{ ..., -2.1, 3, 0.91278393, ... \}$

Describing Sets in Words

Sets can be described using real language to specify their elements.

• Example: $Q$: the set of all provinces in Canada. $Q = \{\text{B.C.}, \text{Ontario}, ...\}$

• Example: $P$: the set of all provinces whose residents do not own a house. $P = \{\text{Ontario}, \text{Alberta}, \text{B.C.}\}$

Elements and Membership

To indicate that an object $x$ is an element of a set $S$, we write $x \in S$. If $x$ is not in $S$, we write $x \notin S$.

• Example: $x \in Q$ means $x$ is a province in Canada.

• Example: $y \notin Q$ means $y$ is not a province in Canada.

UNIVERSAL SET

Definition

The universal set is the set of all elements under consideration in a particular context, denoted by $U$.

• Example: $U$: all students in a school.

• Subsets: $A$: students in grade 9; $B$: students in grade 10; $C$: students in grade 11.

Venn Diagrams

Venn diagrams are graphical representations of sets and their relationships within the universal set.

• Each circle represents a set; the rectangle represents the universal set $U$.

• Overlapping regions show elements common to multiple sets.

SUBSETS

Definition

A set $A$ is a subset of set $B$ if every element of $A$ is also an element of $B$. This is denoted $A \subseteq B$.

• Example: $A = \{\text{fish}\}$, $B = \{\text{fish}, \text{cat}, \text{dog}\}$, so $A \subseteq B$.

• Non-example: $\{\text{humans}\} \not\subseteq \{\text{fish}, \text{cat}, \text{dog}\}$

• Properties: The empty set $\varnothing$ is a subset of every set, and every set is a subset of itself.

Subset Relationships

• If $A \subseteq U$, then $A$ is contained within the universal set $U$.

• Example: $\{\text{lobster}, 100, \heartsuit\} \subseteq C$

SET OPERATIONS

Union

The union of two sets $A$ and $B$, denoted $A \cup B$, is the set of all elements that are in $A$, in $B$, or in both.

• Definition: $A \cup B = \{x : x \in A \text{ or } x \in B\}$

• Example: If $A = \{1, \heartsuit\}$ and $B = \{3, 5\}$, then $C = A \cup B = \{1, \heartsuit, 3, 5\}$

Intersection

The intersection of two sets $A$ and $B$, denoted $A \cap B$, is the set of all elements that are in both $A$ and $B$.

• Definition: $A \cap B = \{x : x \in A \text{ and } x \in B\}$

• Example: If $A = \{1, 3, 5\}$ and $B = \{3, 5, 7\}$, then $A \cap B = \{3, 5\}$

Complement

The complement of a set $A$ (relative to the universal set $U$), denoted $A'$, is the set of all elements in $U$ that are not in $A$.

• Definition: $A' = \{x \in U : x \notin A\}$

• Example: If $U = \{1, 2, 3, 4, 5\}$ and $A = \{2, 4\}$, then $A' = \{1, 3, 5\}$

SIZE OF SETS

Cardinality

The cardinality of a set $A$, denoted $n(A)$, is the number of elements in $A$.

• Example: If $A = \{1, 2, 3\}$, then $n(A) = 3$.

Formulas for Union and Intersection

• Union Formula: For two sets $A$ and $B$, the number of elements in their union is given by: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

• Example Application: If a group of participants could receive therapy A, therapy B, or both, then: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ where $n(A)$ is the number who received therapy A, $n(B)$ is the number who received therapy B, and $n(A \cap B)$ is the number who received both.

Summary Table: Set Operations

Operation	Symbol	Definition	Example
Union	$A \cup B$	All elements in $A$ or $B$	$\{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}$
Intersection	$A \cap B$	Elements in both $A$ and $B$	$\{1, 2\} \cap \{2, 3\} = \{2\}$
Complement	$A'$	Elements in $U$ not in $A$	If $U = \{1, 2, 3\}$, $A = \{2\}$, $A' = \{1, 3\}$
Subset	$A \subseteq B$	All elements of $A$ are in $B$	$\{1\} \subseteq \{1, 2\}$

Additional info:

• Venn diagrams are a useful tool for visualizing relationships between sets, especially for union, intersection, and complement.

• Set theory forms the foundation for probability and statistics, as events can be represented as sets and their relationships analyzed using set operations.