What is Sets

Introduction of Sets

In Maths, sets are a collection of well-defined objects or elements. A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set in a curly bracket {…}.

A Sets is defined as a collection of distinct objects of the same type or class of objects. The purposes of a set are called elements or members of the set. An object can be numbers, alphabets, names, etc. Examples of sets are:

A set of rivers of India.
A set of vowels.

We broadly denote a set by the capital letter A, B, C, etc. while the fundamentals of the set by small letter a, b, x, y, etc.

If A is a set, and a is one of the elements of A, then we denote it as a ∈ A. Here the symbol ∈ means -"Element of."

Sets Representation:

Sets are represented in two forms

1. Roster or tabular form: In this form of representation we list all the elements of the set within braces { } and separate them by commas.

Example: If A= set of all odd numbers less then 10 then in the roster from it can be expressed as A={ 1,3,5,7,9}.

2. Set Builder form: In this form of representation we list the properties fulfilled by all the elements of the set. We note as {x: x satisfies properties P}. and read as 'the set of those entire x such that each x has properties P.'

Example: If B= {2, 4, 8, 16, 32}, then the set builder representation will be: B={x: x=2n, where n ∈ N and 1≤ n ≥5}

Standard Notations

Cardinality of a Sets:

The total number of unique elements in the set is called the cardinality of the set. The cardinality of the countably infinite set is countably infinite.

What are the types of Sets?

A set has many types, such as;

• Empty Set or Null set: It has no element present in it.Example: A = {} is a null set.
• Finite Set: It has a limited number of elements.Example: A = {1,2,3,4}
• Infinite Set: It has an infinite number of elements.Example: A = {x: x is the set of all whole numbers}
• Equal Set: Two sets which have the same members.Example: A = {1,2,5} and B={2,5,1}: Set A = Set B
• Subsets: A set ‘A’ is said to be a subset of B if each element of A is also an element of B.Example: A={1,2}, B={1,2,3,4}, then A ⊆ B
• Universal Set: A set which consists of all elements of other sets present in a Venn diagram.Example: A={1,2}, B={2,3}, The universal set here will be, U = {1, 2,3}

Sets Formulas

What is a Power set? Define with example.

Answer: In set theory, the power set of a set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set.

Let A and B be two finite sets such that n(A) = 25, n(B) = 20 and n(A ∪ B) = 30, find n(A ∩ B).

If n(A – B) = 30, n(A ∪ B) = 65 and n(A ∩ B) = 22, then find n(B).