Sets

Understanding Sets

• A set is a collection of distinct objects or elements.
• Elements in a set are represented within curly brackets { }.
• The order in which elements appear does not affect the nature of the set.
• Objects that belong to a set are called its members or elements.
• A set that does not contain any element is called an empty set or null set, represented by { } or ∅.
• Two sets that contain exactly the same elements, irrespective of their order, are considered equal.

Types of Sets

• Finite Set: A set with a countable number of elements. For example, the set of all English alphabets, A = {a, b, c, …, z}.
• Infinite Set: A set with countless elements. For example, the set of all natural numbers, N = {1, 2, 3, …}.
• Equivalent Set: Two sets with the same number of elements. For example, A = {1, 2, 3}, B = {a, b, c} are equivalent sets.
• Subset: A set ‘A’ is a subset of set ‘B’ if all elements of set ‘A’ are also elements of ‘B’. This is symbolised as A ⊆ B.

Operations on Sets

• Union of sets (A ∪ B): It is a set consisting of all elements that are in A or B or in both.
• Intersection of sets (A ∩ B): It is a set consisting of all elements that are common to both A and B.
• Difference of sets (A - B): It is a set consisting of elements present in A but not in B.
• Complement of a set (‘A’): It is a set consisting of elements not present in ‘A’ but are in the universal set.

Venn Diagrams

• A Venn diagram is a pictorial way of representing sets and their relationships.
• Each set is represented by a circle or other shape.
• The universal set is represented by a rectangle enclosing all other sets.
• Where circles overlap, this signifies the intersection of sets (i.e., common elements).
• A region in the rectangle, but outside of any circle, signifies elements of the universal set that are not members of any other set.