[size=150]Set theory is a foundational branch of mathematics that deals with the study of sets, which are collections of distinct objects. In the context of discrete mathematics, set theory provides a formal framework for understanding and reasoning about collections of elements.[br][br]In set theory, the fundamental concept is a set, which is an unordered collection of unique objects, called elements or members. These elements can be anything: numbers, letters, symbols, or even other sets. For example, a set could be a collection of integers, a set of colors, or a set of students in a class.[br][br][u][size=200]Key Concepts in Set Theory:[/size][/u][br][br]1.[b][u] Elements:[/u][/b] The objects or entities that belong to a set. Each element is distinct and appears only once in a set.[br][br]2.[b][u] Set Notation:[/u][/b] Sets are typically denoted using curly braces {}. For example, {1, 2, 3} represents a set containing the elements 1, 2, and 3.[br][br]3.[b][u] Cardinality:[/u][/b] The cardinality of a set refers to the number of elements it contains. It is denoted by |A|, where A is the set. For example, if A = {1, 2, 3}, then |A| = 3.[br][br]4.[b][u] Subset[/u][/b]: A set A is said to be a subset of another set B if every element of A is also an element of B. It is denoted as A ⊆ B. If A is a subset of B, but not equal to B, it is called a proper subset and is denoted as A ⊂ B.[br][br]5. [b][u]Union and Intersection:[/u][/b] The union of two sets A and B, denoted by A ∪ B, is the set that contains all the elements present in either A or B. The intersection of two sets A and B, denoted by A ∩ B, is the set that contains only the elements common to both A and B.[br][br]6.[b][u] Complement:[/u][/b] The complement of a set A, denoted by A', is the set of all elements in the universal set U that are not in A.[br][br]7.[b][u] Cartesian Product[/u][/b]: The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. It represents all possible combinations of elements from A and B.[br][br]Set theory provides a foundation for many other branches of mathematics and is widely used in various areas, including logic, algebra, analysis, and computer science. It offers a rigorous framework for defining and manipulating collections of objects, allowing for precise reasoning and formalization of mathematical concepts..[/size][br]