Sets are fundamental building blocks of mathematics and data science. They allow us to organize, group, and analyze data effectively. From defining unique elements to performing operations like union and intersection, sets provide a versatile tool for structuring and analyzing data.

Sets Overview

A set is a collection of distinct objects, called elements, grouped together. Sets are denoted using curly braces, such as:

Key notation:

• If $x$ is an element of set $A$, we write $x \in A$.
• If $x$ is not in $A$, we write $x \notin A$.

Types of Sets

• Finite sets: sets with a limited number of elements; $$A = \{2, 4, 6, 8\}$$
• Infinite sets: sets with an infinite number of elements; $$\mathbb{N} = \{1, 2, 3, ...\}$$
• Empty sets: sets with no elements, denoted by $\emptyset$; $$A = \emptyset$$
• Subsets: a set $A$ is a subset of $B$ if all elements of $A$ are in $B$; $$A = \{1, 2\},\ B = \{1, 2, 3\},\ A \subseteq B$$
• Universal sets: the set containing all possible elements in a particular context, denoted $U$; $$U = \{\text{All integers}\}$$
• Power sets: the set of all subsets of a set. $$P(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$$

Set Operations

Sets enable several operations to compare and manipulate data. Some key operations include (for $A = \{1,2\},\ B = \{2,3\}$):

• Union: combines elements from sets $A$ and $B$; $$A \cup B = \{1,2,3\}$$
• Intersection: finds common elements between sets $A$ and $B$; $$A \cap B = \{2\}$$
• Difference: elements in $A$ but not in $B$; $$A - B = \{1\}$$
• Complement: elements not in $A$ but in the universal set $U$; $$A' = U - A$$
• Cartesian product: the set of all ordered pairs between sets $A$ and $B$. $$A \times B = \{(1,2), (1,3), (2,2), (2,3)\}$$

Real-World Applications

Sets are crucial for solving problems in data science and analytics:

• Data organization: grouping unique items (e.g., distinct customer IDs);
• Data cleaning: removing duplicate entries using set properties;
• Set operations: finding intersections (common features) or differences (unique features) in datasets;
• Probability: computing union or intersection of events;
• Database queries: using sets to perform operations like joins, unions, and differences.