Sets and Set Operations

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics.

Definition of a Set

A set is a collection of objects, called the elements or members of the set. A set is defined by specifying the properties that its members must satisfy, or by listing its members between curly braces.

For example, a set of natural numbers less than 10 can be written as:

S = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Basic Set Operations

  1. Union of sets (A ∪ B): The union of two sets A and B is the set of elements which are in A, in B, or in both.

  2. Intersection of sets (A ∩ B): The intersection of two sets A and B is the set of elements which are in both A and B.

  3. Difference of sets (A - B): The difference of the set B from the set A (also known as the set-theoretic difference of A and B), is the set of elements that are in A but not in B.

  4. Complement of a set (A): The complement of a set A refers to elements not in A.

  5. Cartesian product (A x B): The Cartesian product of A and B is the set of all ordered pairs (a, b) where a is in A and b is in B.

These operations can be used to manipulate sets in various ways, and they form the basis for a branch of mathematics known as set theory.