Introduction
Set theory is a fundamental branch of mathematics that studies sets, which are collections of objects. It serves as a foundational system for nearly all mathematical concepts.
Basic Definitions and Notation
A set is a well-defined collection of distinct objects. The objects are called elements or members.
- $x \in A$: $x$ is an element of set $A$.
- $x \notin A$: $x$ is not an element of set $A$.
- $\emptyset$ or $\{\}$: the empty set (no elements).
Examples:
- $A = \{1, 2, 3, 4\}$ (set of first four natural numbers)
- $B = \{x \mid x \text{ is an even prime number}\} = \{2\}$
- $\mathbb{N} = \{1, 2, 3, \dots\}$ (natural numbers)
- $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ (integers)
Set Operations
Let $A$ and $B$ be sets. The following operations are defined:
| Operation | Symbol | Definition |
|---|---|---|
| Union | $A \cup B$ | $\{x \mid x \in A \text{ or } x \in B\}$ |
| Intersection | $A \cap B$ | $\{x \mid x \in A \text{ and } x \in B\}$ |
| Difference | $A \setminus B$ | $\{x \mid x \in A \text{ and } x \notin B\}$ |
| Complement | $A^c$ | $\{x \in U \mid x \notin A\}$ (relative to universal set $U$) |
Example
Let $U = \{1,2,3,4,5,6\}$, $A = \{1,2,3\}$, $B = \{3,4,5\}$.
$A \cup B = \{1,2,3,4,5\}$
$A \cap B = \{3\}$
$A \setminus B = \{1,2\}$
$A^c = \{4,5,6\}$
Venn Diagrams (Figures)
Venn diagrams visually represent sets and their relationships.
Subsets and Power Sets
- $A \subseteq B$ means every element of $A$ is also an element of $B$.
- $A \subset B$ means $A \subseteq B$ and $A \neq B$ (proper subset).
- The power set of $A$, denoted $\mathcal{P}(A)$, is the set of all subsets of $A$.
Example:
If $A = \{1, 2\}$, then
$\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}$
Cardinality: $2^{|A|} = 2^2 = 4$
Cardinality
The cardinality of a set $A$, denoted $|A|$, is the number of elements in $A$.
- $|\{a,b,c\}| = 3$
- $|\emptyset| = 0$
- $|\mathbb{N}| = \infty$ (countably infinite)
Set Identities (Laws)
| Law | Expression |
|---|---|
| Commutative | $A \cup B = B \cup A$, $A \cap B = B \cap A$ |
| Associative | $(A \cup B) \cup C = A \cup (B \cup C)$ |
| Distributive | $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ |
| De Morgan's | $(A \cup B)^c = A^c \cap B^c$, $(A \cap B)^c = A^c \cup B^c$ |
| Identity | $A \cup \emptyset = A$, $A \cap U = A$ |
| Complement | $A \cup A^c = U$, $A \cap A^c = \emptyset$ |
Cartesian Product
The Cartesian product of $A$ and $B$ is:
$A \times B = \{(a,b) \mid a \in A, b \in B\}$
Example:
$A = \{1,2\}$, $B = \{x,y\}$:
$A \times B = \{(1,x), (1,y), (2,x), (2,y)\}$
Conclusion
Set theory provides the language and basic constructs for nearly all areas of mathematics, including relations, functions, and number systems. Understanding sets, operations, Venn diagrams, and cardinality is essential for advanced mathematical study.
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