Introduction to Set Theory and Venn Diagrams
Set theory is the mathematical study of collections of objects. It forms the foundation of modern mathematics and is essential for understanding probability, logic, and abstract algebra.
1. Fundamental Definitions
What is a Set?
A set is a well-defined collection of distinct objects, called elements or members. Sets are typically denoted by capital letters ($A$, $B$, $C$, ...) and their elements are enclosed in curly braces.
Example: $A = \{1, 2, 3, 4, 5\}$ is the set containing the first five positive integers.
Notation
- $x \in A$ means "$x$ is an element of $A$"
- $x \notin A$ means "$x$ is not an element of $A$"
- $|A|$ or $n(A)$ denotes the cardinality (number of elements) of $A$
- $\emptyset$ or $\{\}$ denotes the empty set
- $U$ or $\xi$ denotes the universal set
Ways to Define a Set
1. Roster (Listing) Method:
$B = \{2, 4, 6, 8, 10\}$
$B = \{2, 4, 6, 8, 10\}$
2. Set-Builder Notation:
$B = \{x \mid x \text{ is an even positive integer}, x \leq 10\}$
Read: "the set of all $x$ such that $x$ is an even positive integer less than or equal to 10"
$B = \{x \mid x \text{ is an even positive integer}, x \leq 10\}$
Read: "the set of all $x$ such that $x$ is an even positive integer less than or equal to 10"
3. Description Method:
$B$ = the set of even positive integers up to 10
$B$ = the set of even positive integers up to 10
Special Sets
- $\mathbb{N}$ = Natural numbers $\{1, 2, 3, ...\}$
- $\mathbb{Z}$ = Integers $\{..., -2, -1, 0, 1, 2, ...\}$
- $\mathbb{Q}$ = Rational numbers
- $\mathbb{R}$ = Real numbers
2. Subsets and Set Relationships
Subset ($\subseteq$)
Set $A$ is a subset of set $B$ if every element of $A$ is also an element of $B$:
$$A \subseteq B \iff (\forall x)(x \in A \Rightarrow x \in B)$$
Proper Subset ($\subset$)
$A$ is a proper subset of $B$ if $A \subseteq B$ and $A \neq B$.
Key Facts:
- $\emptyset \subseteq A$ for every set $A$
- $A \subseteq A$ for every set $A$
- If $|A| = n$, then $A$ has $2^n$ subsets
Equal Sets
Two sets are equal if and only if they contain exactly the same elements:
$$A = B \iff A \subseteq B \text{ and } B \subseteq A$$
Example:
Let $A = \{1, 2, 3\}$ and $B = \{1, 2, 3, 4, 5\}$
Then $A \subset B$ (proper subset)
$n(A) = 3$, so $A$ has $2^3 = 8$ subsets:
$\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}$
Let $A = \{1, 2, 3\}$ and $B = \{1, 2, 3, 4, 5\}$
Then $A \subset B$ (proper subset)
$n(A) = 3$, so $A$ has $2^3 = 8$ subsets:
$\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}$