Sets Class 11 Notes CBSE Maths Chapter 1 (Free PDF Download)

Section–A (1 Mark Questions)

1. In a town of 840 persons, 450 persons read Hindi, 300 read English and 200 read both. Then find the number of persons who read neither.

Ans. Let $\mathrm{H}$ be the set of persons who read Hindi and $\mathrm{E}$ be the set of persons who read English.

Then, $n(U)=840, n(H)=450$,

$$ n(E)=300, n(H \cap E)=200 $$

Number of persons who read neither

$$ \begin{aligned} & =n\left(H ' \cap F^{\prime}\right)=n\left((H \cup E)^{\prime}\right) \\ & =n(U)-n(H \cup E) \\ & =840-[n(H)+n(E)-n(H \cap E)] \\ & =840-(450+300-200)=290 \end{aligned} $$

2. If X and Y are two sets and X’ denotes the complement of X, then $X\cap (X\cap Y){}'$is equal to ------------.

Ans.

$ \begin{aligned} & \text { Let } x \in X \cap(X \cup Y)^{\prime} \\ & \Rightarrow x \in X \cap\left(X Y^{\prime} \cap Y^{\prime}\right) \\ & \Rightarrow x \in\left(X \cap X^{\prime}\right) \cap\left(X \cap Y^{\prime}\right) \\ & \Rightarrow x \in \varnothing \cap\left(X \cap Y^{\prime}\right) \quad\left[\because A \cap A^{\prime}=\varnothing\right] \\ & \Rightarrow x \in \varnothing \quad \end{aligned} $

3. If A is any set, then $A\subset A$. State True or false.

Ans. Since every set is a subset of itself. So, it is 'True'.

4. State True/False. The sets {1,2,3,4} and (3,4,5,6} are equal.

Ans. False, since the two sets do not contain the same elements.

5. Given A={0,1,2}, $B=\left \{ x\epsilon R|0\leq x\leq 2 \right \}$then is A=B.

Ans. False, Here $A=\{0,1,2\}, B$ is a set having all real numbers from 0 to 2

So $A \neq B$. Hence, the given statement is 'False'.

Section-B (2 Marks Questions)

6. If $X=\left \{ 8^{n}-7n-1|\;n\epsilon N \right \}$ and Y=\left \{ 49n-49|\;n\epsilon N \right \}, the $Y\subset X$. Yes or No?

Ans. $X=\left\{8^n-7 n-1 \mid n \in N\right\}=\{0,49,490, \ldots\}$

$$Y=\{49 n-49 \mid n \in N\}=\{0,49,98,147, \ldots, 490, \ldots\}$$

Clearly, every element of $X$ is in $Y$ but every element of $Y$ is not in $X$.

$$\therefore \quad X \subset Y$$

7. Write the power set of each of the following sets:

1. $A=\left \{ x;x\epsilon R\;and\;x^{2}+7=0 \right \}$

2. $B=\left \{ y;y\epsilon N\;and\;1\leq y\leq 3 \right \}$

Ans. (a) Clearly, $A=\varnothing$ (Null set)

$\therefore \varnothing$ is the only subset of given set.

$\therefore P(A)=\{\varnothing\}$

(b) The set $\mathrm{B}$ can be written as $\{1,2,3\}$

Subsets of B are

$$ \begin{aligned} & \varnothing,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\} \\ & \therefore P(B)=\{\varnothing,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\} . \end{aligned} $$

8.  If Y = {x | x } is a positive factor of the number $2^{p}(2^{p}-1)$ where $2^{p}-1$

is a prime number. Write Y in roster form.

Ans. $\mathrm{Y}=\{\mathrm{X} \mid \mathrm{x}$ is a positive factor of the number

$2^p\left(2^p-1\right)$, where $2^p-1$ is a prime number\}.

So, the factors of $2^p$ are $1,2,2^2, 2^3, \ldots, 2^p$.

$Y=\left\{1,2,2^2, 2^3, \ldots, 2^p,\left(2^p-1\right)\right\}$

9. Given L={1,2,3,4}, M={3,4,5,6} and N={1,3,5}, Verify that $L-(M\cup N)=(L-M)\cap (L-N)$ 

Ans. Given

$$\begin{aligned}& L=\{1,2,3,4\}, M=\{3,4,5,6\} \text { and } N=\{1,3,5\} \\& M \cup N=\{1,3,4,5,6\}, L-(M \cup N)=\{2\}\end{aligned}$$

Now, $L-M=\{1,2\}$ and $L-N=\{2,4\}$

$$(L-M) \cap(L-N)=(2)$$

Hence, $L-(M \cup N)=(L-M) \cap(L-N)$.

10. Use the properties of sets to prove that for all the sets A and B , $A-(A \cap B)=A \cap(A \cap B)$ 

Ans. We have, $A-(A \cap B)=A \cap(A \cap B) '$

(Since $A-B=A \cap B^{\prime}$ )

$=A \cap\left(A \cup B^{\prime}\right) \quad$ (by De Morgan's law)

$=\left(A \cap A^{\prime}\right) \cup\left(A \cap B^{\prime}\right) \quad$ (by distributive law)

$=\phi \cup\left(A \cap B^{\prime}\right)$

$=A \cap B^{\prime}=A-B$

11. Let R and S be the sets defined as follows:

$R = \left \{ x\epsilon Z\;|\; x\; is\;divisible\;by\;2 \right \}$

$S = \left \{ y\epsilon Z\;|\; y\; is\;divisible\;by\;3 \right \},\;then\;R\cap S= \varnothing $

Ans. False

Since 6 is divisible by both 3 and 2 .

Thus, $R \cap S \neq \varnothing$

12. A, B and C are subsets of Universal set. If A = {2, 4, 6, 8, 12, 20}, B = {3, 6, 9, 12, 15}, C = {5, 10, 15, 20} and U is the set of all whole numbers, draw a Venn diagram showing the relation of U, A, B and C.

PDF Summary - Class 11 Maths Sets Notes (Chapter 1)

Introduction to Sets

The concept of set acts as an important part of present-day Mathematics. This concept is almost used in each and every branch of Mathematics. Sets are generally used to define the concept of relations and functions in Mathematics. The study of different Mathematics topics including Geometry, Probability, and Sequence requires thorough knowledge of sets.

German Mathematician George Cantor introduced the theory of sets in Mathematics. He observed the concept Set while working on “Problems On Trigonometric Series”.

What is Set in Mathematics?

In Mathematics, a Set is defined as the well-defined collection of objects or numbers. Each member that belongs to the set is termed as the element of the set. All members or elements of the set are unique. For example: {1, 2, 3, 4, 5} is the set of counting numbers less than 6.

Set Notation 

The notation of the set is fairly simple. Each element or member in a set is separated by a comma, and then curly brackets are placed around the whole thing. Look at the image below to understand the set notation precisely.

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Image: Set Notation

Curly brackets are also known as “set brackets” or “braces”.

Set Notation Example

Set N is a collection of all the natural numbers.

Set N is represented in the following way:

N = { 1, 2, 3, 4, 5…}

The three dots in the above set are known as “Ellipse” or “Continue-On''.

Representation of Sets

Sets are represented in two forms:

1. Roster or Tabular Form: In roaster form, all the elements of a set are separated by a comma and are enclosed with braces {}. For example, the set of all even positive integers less than 5 is represented as { 2, 4}.

2. Set Builder Form:  In this form, all the elements belonging to a set have a common property.  For example, in the set { a, e, i, o, u}, all the elements of this set have a common property. Each of the alphabets in the set is a vowel, and no other set possesses this same property.