Skip to content
← Applied Mathematics

Applied Mathematics · Class 11 Commerce

Ch 3Sets — Class 11 Applied Mathematics, concept-first.

This concept map shows how the chapter fits together. A set can be represented in roster or set-builder form; sets come in several types (empty, finite, infinite, subset, universal — with intervals as subsets of the reals); the operations on sets are union, intersection, complement and De Morgan's laws; and Venn diagra…

49

Q&A

9

Concepts

Not available

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

What Is a Set?

Imagine you have a handful of marbles. You can put them together in a little pile and say "these are my marbles." That pile — that collection — is a set. The marbles themselves are the elements (or members) of the set.

Most relevant Q&A

Start with this concept →

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Concept Map

This concept map shows how the chapter fits together. A set can be represented in roster or set-builder form; sets come in several types (empty, finite, infinite, subset, universal — with intervals as…

Introduction

Every day we talk casually about 'collections' — the books on a shelf, the songs on a favourite playlist, the rules a school expects everyone to follow.

3.1

What Is a Set?

A set is a well-defined collection of distinct objects — 'well-defined' meaning there is a clear rule that lets us decide, for any object, whether it belongs to the collection or not.

+Worked Examplesi1 question
  1. Example 1Which of the following collections is not a set? Give reason for your answers. (i) The collection of all boys of your class. (ii) The collec…Preview
3.2

Representation of Sets

7 Q

Sets are abstract collections, but to work with them we need a concrete way to write them down. The most direct method is the roster or tabular form, where we simply list every element inside curly br…

+Worked Examplesi2 questions
  1. Example 2Write the following sets in roster form: (i) $A = \{x : x \text{ is a letter in the word 'MATHEMATICS'}\}$ (ii) $B = \{x : x \text{ is a let…Free
  2. Example 3Write the following sets in the set-builder form: (i) $A = \{1, 4, 9, 16\}$ (ii) $B = \{2, 3, 5, 7, 11\}$ (iii) $C = \{-2, 0, 2\}$Preview
+Exercise 3.1i5 questions
  1. Q1Which of the following are sets? (i) The collection of most talented authors of India. (ii) The collection of all months of a year beginning…Free
  2. Q2Write the following in set-builder form: (i) $A = \{4, 8, 12, 16, 20\}$ (ii) $B = \{2, 3, 5, 7, 11, 13, 17, 19, \ldots\}$ (iii) $C = \{b, c,…Free
  3. Q3Write the following in roster form: (i) $A = \{x : x \text{ is a whole number less than } 5\}$ (ii) $B = \{x : x \text{ is an integer and }…Preview
  4. Q4Match each of the sets on the left expressed in roster form with the same set described in the set builder form on the right: (i) $\{1, 2, 3…Preview
  5. Q5Let $A = \{1, 2, 3, 4, 5\}$, $B = \{x : x \text{ is a factor of } 4\}$, $C = \{1, 4, 9\}$. Insert the correct symbol '$\in$' or '$\notin$' i…Preview
3.3

Types of Sets

Sets are not all the same — they come in different flavours depending on how many elements they hold and how they relate to each other.

+Worked Examplesi1 question
  1. Example 4State which of the following sets are finite and which are infinite. (i) $\{x : x \in Z \text{ and satisfies the equation } (x+1)(x+2) = 0\}…Preview
3.4

Subsets

We now turn to a natural and powerful idea: when every element of one set is also contained in another, we say the first is a subset of the second.

+Worked Examplesi2 questions
  1. Example 5Consider the sets $\phi$; $A = \{1, 2, 3\}$; $B = \{1, 4, 5\}$; $C = \{1, 2, 3, 4, 5\}$; $D = \{x : x \in N \text{ and } x < 4\}$. Insert th…Free
  2. Example 6Let $A = \{1, 2, 3, \{4\}, 5\}$. Which of the following are incorrect? Give reasons. (i) $1 \in A$ (ii) $\{1, 2\} \subseteq A$ (iii) $\{1, 2…Preview
3.5

Power Set

The idea of a set is to collect objects, but what if we turn the collection itself into an object? The power set does exactly that — it is the set of all possible subsets of a given set.

3.6

Universal Set

We have seen how sets can be combined and compared, but every discussion in set theory takes place within a fixed, agreed-upon "universe" of objects.

3.7

Intervals as Subset of the Set of Real Numbers R

The real number line is not just a collection of isolated points — it is a continuous stretch. An interval is a connected block of that line, containing every real number that lies between two given e…

3.8

Types of Intervals

11 Q

An interval is simply a connected block of real numbers. The key idea is that we need a precise way to describe whether the endpoints of that block are included or excluded.

+Worked Examplesi4 questions
  1. Example 7The interval notation for all real numbers from $-3$ to $2$, as a closed interval, is written as $[-3, 2] = \{x : x \in R, -3 \leq x \leq 2\…Free
  2. Example 8The interval of real numbers between $-3$ and $2$ (excluding both end points) is written as the open interval $(-3, 2) = \{x : x \in R, -3 <…Free
  3. Example 9Consider the interval $(-3, 2]$, where $(-3, 2] = \{x : x \in R, -3 < x \leq 2\}$ represents the set of real numbers between $-3$ and $2$ wh…Preview
  4. Example 10Find the length of the interval $(2, 7)$ and the length of the interval $(-3, 4)$.Preview
+Exercise 3.2i7 questions
  1. Q1Which of the following sets are finite and which are infinite? In case of finite sets, write its cardinality. (i) $\{x : x \text{ is a natur…Free
  2. Q2Which of the following sets are empty and which are singleton sets? (i) $\{x : x \text{ is an even prime number}\}$. (ii) $\{x : x \text{ is…Free
  3. Q3Which of the following pairs of sets are equal? Give reasons. (i) $A = \{-2, 3\}$, $B = \{x : x \text{ is a solution of } x^2 - x - 6 = 0\}$…Free
  4. Q4Let $A = \{1, 2, \{3, 4\}, 5\}$. Put the correct symbol in each of the following. (i) $\{3, 4\}$ ___ $A$ (ii) $\{1\}$ ___ $A$ (iii) $\{3\}$…Preview
  5. Q5Write the following in set-builder form. (i) $(1, 3)$ (ii) $(-1, 3)$ (iii) $(-4, 0)$ (iv) $(-1, 1)$ (v) $(0, \infty)$Preview
  6. Q6Let $A = \phi$. Find $P(P(A))$.Preview
  7. Q7Find the number of subsets of the set $B = \{a, b, c, d\}$.Preview
3.9

Venn Diagrams

Venn diagrams turn the abstract language of sets into a picture you can see and reason with. Instead of juggling symbols like , , and in your head, you draw overlapping circles inside a rectangle (the…

+Worked Examplesi3 questions
  1. Example 11Let $U = \{1, 2, 3, 4, 5\}$ and $A = \{2, 4\}$. Represent the relationship between $U$ and $A$ using a Venn diagram.Free
  2. Example 12Let $U = \{1, 2, 3, 4, 5\}$, $A = \{1, 2\}$, $B = \{3, 5\}$. Represent the relationship between $U$, $A$ and $B$ using Venn diagrams.Preview
  3. Example 13Let $U = \{1, 2, 3, 4, 5\}$, $A = \{1, 2, 3\}$ and $B = \{1, 2\}$. Represent the relationship between $U$, $A$ and $B$ using Venn diagrams.Preview
3.10

Operations on Sets

Sets are the building blocks of mathematics, and just as we add or multiply numbers, we can combine and compare sets using special operations.

+Exercise 3.3i9 questions
  1. Q1For the following sets, find their union and intersection. (i) $A = \{x : x \text{ is the letter of the word 'MATHEMATICS'}\}$. $B = \{x : x…Free
  2. Q2Let $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$; $A = \{1, 2, 3, 4\}$; $B = \{3, 4, 6\}$; $C = \{5, 6, 7, 8\}$, find: (i) $A - (B \cup C)$ (ii) $A \cap…Free
  3. Q3Let $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$; $A = \{1, 2, 3\}$; $B = \{2, 4, 6\}$, $C = \{3, 6, 9, 12\}$. Verify: (i) $(A \cup B)' = A' \cap B'$ (i…Free
  4. Q4Draw suitable Venn diagrams for each of the following. (i) $(A \cup B)'$ (ii) $(A \cap B)'$ (iii) $A' \cap B'$ (iv) $A' \cup B'$Preview
  5. Q5Two finite sets have 'm' and 'n' elements. The total number of subsets of the first set is 56 more than the total number of subsets of the s…Preview
  6. Q6Let $A = \{a, b, c\}$ and $B = \{a, b, c, d\}$. Is $A \subseteq B$? What is $A \cup B$? What is $A \cap B$?Preview
  7. Q7Fill in the blanks. (i) $A' \cap \phi = $ ___ (ii) $A \cap \phi' = $ ___ (iii) $A \cup U' = $ ___ (iv) $A \cap U' = $ ___ (v) $A \cap \phi'…Preview
  8. Q8If $A = \{1, 2, 3, 4\}$, then the number of subsets of set $A$ containing element $3$, is: (i) 24 (ii) 28 (iii) 8 (iv) 16Preview
  9. Q9If $A = \{1, 2, 3, 4, 5\}$, $B = \{2, 4, 6\}$ and $C = \{3, 4, 6\}$, then $(A \cup B) \cap C$ is (i) $\{3, 4, 6\}$ (ii) $\{1, 2, 3\}$ (iii)…Preview
3.11

Practical Problems on Operations on Sets

15 Q

Practical problems on sets are where the abstract operations of union, intersection, and complement meet real-world counting.

+Worked Examplesi10 questions
  1. Example 14Out of 20 members in a family, 12 like tea and 15 like coffee. Assume that each one likes atleast one of the two drinks, how many like: (i)…Free
  2. Example 15In a group of 400 people, who speak either Hindi or English or both, if 230 speak Hindi only and 70 speak both English and Hindi, then how m…Free
  3. Example 16In a group of 70 people, 45 speak Hindi language and 33 speak English language and 10 speak neither Hindi nor English. How many can speak bo…Free
  4. Example 17In a certain town, 25% of the families own a phone, 15% own a car, and 65% families own neither a phone nor a car. 2000 families own both a…Preview
  5. Example 18Out of 500 car owners investigated, 400 owned car A and 200 owned car B, 50 owned both A and B cars. Is the data correct?Preview
  6. Example 19Let $U$ be the universal set for sets $A$ and $B$ such that $n(A) = 200$, $n(B) = 300$ and $n(A \cap B) = 100$. Then, $n(A' \cap B') = 300$,…Preview
  7. Example 20A market research group conducted a survey of 1000 consumers and reported that 720 consumers like product A and 450 consumers like product B…Preview
  8. Example 21In a survey of 60 people, it was found that 25 people read newspaper H; 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 re…Preview
  9. Example 22In a class of 35 students, 17 have taken Mathematics, 10 have taken Mathematics but not Economics. If each student has taken either of the t…Preview
  10. Example 23With corona virus threating to run riot in India, prevention appears to be the best cure available so far. It is crucial for people to have…Preview
+Exercise 3.4i5 questions
  1. Q1In a group of 70 people, 37 like coffee, 52 like tea and each person likes atleast one of the two drinks. How many people like both coffee a…Free
  2. Q2In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only? How many like tennis.Free
  3. Q3In a survey of 600 students in a school, 150 students were found to be taking apple juice, 225 taking orange juice and 100 were taking both…Preview
  4. Q4In an election, two contestants A and B contested. $y\%$ of the total votes voted for A and $(y+30)\%$ for B. If 20% of the voters did not v…Preview
  5. Q5In a class, 70 students wrote two tests, test-I and test-II. 50% of the students failed in test-I and 40% of the students in test-II. How ma…Preview