Introduction to Set Theory

A set is a well-defined collection of distinct objects, called elements or members. In Mathematics, sets are denoted by capital letters (e.g., A, B, C) and elements are listed within curly brackets {}.

Types of Sets

• Empty (Null) Set: A set with no elements, denoted by ∅ or {}.
• Universal Set (U): The set containing all possible elements under consideration.
• Finite Set: A set whose elements can be counted (e.g., factors of 6).
• Infinite Set: A set where counting elements never ends (e.g., set of all prime numbers).
• Subset (⊂): If every element of set A is also in set B, A is a subset of B.
• Disjoint Sets: Sets with no common elements (Intersection is empty).
• Complement (A'): Elements in the Universal set that are not in set A.

Algebra of Sets

Common operations include:

• Union (A ⊪ B): Elements in A, or B, or both.
• Intersection (A ∩ B): Elements common to both A and B.
• Difference (A - B): Elements in A that are not in B.

Cardinality of Sets

Cardinality refers to the number of elements in a set, denoted as n(A). For two sets A and B:
n(A ⊪ B) = n(A) + n(B) - n(A ∩ B)

Venn Diagrams

Venn diagrams are pictorial representations of sets. Circles represent sets, and the surrounding rectangle represents the Universal set. For 3 sets (A, B, and C), the formula for the union is:
n(A ⊪ B ⊪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)