NCERT Class 11-Math՚s Exemplar Chapter 1 Sets CBSE Board Problems (For CBSE, ICSE, IAS, NET, NRA 2023)

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Overview

This chapter deals with the concept of a set, operations on sets. Concept of sets will be useful in studying the relations and functions.

Set and their representations: A set is a well-defined collection of objects. There are two methods of representing a set

(i) Roaster or tabular form

(ii) Set builder form

The empty set:

A set which does not contain any element is called the empty set or the void set or null set and is denoted by {} or .

Finite and infinite sets:

A set which consists of a finite number of elements is called a finite set otherwise, the set is called an infinite set.

Subsets:

A set A is said to be a subset of set B if every element of A is also an element of B. In symbols we write .

We denote

Set of real numbers by R

Set of natural numbers by N

Set of integers by Z

Set of rational numbers by Q

Set of irrational numbers by T

We observe that

Equal sets:

Given two sets A and B, if every elements of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal. The two equal sets will have exactly the same elements.

Intervals as subsets of R: Let and . Then

(a) An open interval denoted by (a, b) is the set of real numbers

(b) A closed interval denoted by [a, b] is the set of real numbers

(c) Intervals closed at one end and open at the other are given by

Power set:

The collection of all subsets of a set A is called the power set of A. It is denoted by . If the number of elements in A = n , i.e.. , n (A) = n, then the number of elements in .

Universal set

This is a basic set; in a particular context whose elements and subsets are relevant to that particular context. For example, for the set of vowels in English alphabet, the universal set can be the set of all alphabets in English. Universal set is denoted by U.

Venn Diagrams

Venn Diagrams are the diagrams which represent the relationship between sets. For example, the set of natural numbers is a subset of set of whole numbers which is a subset of integers. We can represent this relationship through Venn diagram in the following way.

Union of Sets:

The union of any two given sets A and B is the set C which consists of all those elements which are either in A or in B. In symbols, we write

Some properties of the operation of union.

(i)

(ii)

(iii)

(iv)

(v)