Solve the following Question.(1 Marks)

Question 511 Mark

Write the number of elements in tha power set of null set.

Answer

If A is the void set $\phi$, then P(A) has just one element $\phi$ i.e. $\text{P}(\phi) =\{\phi\}.$

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Question 521 Mark

Write down the all possible subset of the given set.
${1, {1}}.$

Answer

The set has $2$ element, so power set has $2^2 = 4$ elemente, namely, $\phi,$ ${1}, {{1}}, {1, {1}}.$

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Question 531 Mark

The given collections are sets or not? Justify your answer:
The collection of all months of a year beginning with the letter J.

Answer

It forms a set as it is well defined.

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Question 541 Mark

The given set is the example of an empty set or not?
Set of all even natural numbers divisible by 5.

Answer

This set is non-empty as 10 is an even natural number divisible by 5.

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Question 551 Mark

From the sets given below, select equal sets and equivalent sets.
A = {0, a},
B = {1, 2, 3, 4},
C = {4, 8, 12},
D = {3, 1, 2, 4},
E = {1, 0},
F = {8, 4, 12},
G = {1, 5, 7, 11},
H = {a, b}.

Answer

Equal sets:

B and D, because every element of B is a member of D &every element of D is a member of B.
C and F, because every element of C is a member of F & every element of F is a member of C.

Equivalent sets:

(a)	A, E and H	{$\because$ n(A) = n(E) = n(H) = 2}
(b)	B, D and G	{$\because$ n(B) = n(D) = n(G) = 4}
(c)	C and F	{$\because$ n(C) = n(F) = 3}.

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Question 561 Mark

The given following sets are finite & in which of it infinite in if?
$\{\text{x}\in\text{N}:\text{x}>5\}$

Answer

Finite, $\because\{\text{x}\in\text{N}:\text{x}>5\}$ = {6, 7, 8,...} which is infinite.

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Question 571 Mark

If A and B are two sets such that n(A) = 20, n(B) = 25 and $\text{n(A}\cup\text{B)}=40,$ then write $\text{n(A}\cap\text{B)}.$

Answer

$\text{n(A}\cup\text{B)}=\text{n(A) + n(B) - n(A}\cap\text{B)}$
$\text{n(A}\cap\text{B)}=\text{n(A) + n(B) - n(A}\cup\text{B)}$
$=20+25-40$
$=5.$

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Question 581 Mark

The set of all positive integers whose cube is odd.

Answer

As the cube of an odd integer is odd, and an odd positive integer has the form 2n + 1 for same $\text{n}\ge0,$
Hence the set of all positive integers whose cube is odd may be written in set builder form as $\{\text{x}\in \text{Z},\text{x = 2n+1},\text{n}\ge0\}.$

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Question 591 Mark

Describe the following sets in Roster form:
$\{\text{x}\in\text{N}:\text{x}^2 < 25\};$

Answer

In Roster form, we describe a set by listing its elements, separated by commas and the elements are written within braces { }. If a set has infinitely many elements, them comma is followed by ..., where the dots stand for 'and so on'.
$1\in \text{N}\because {1}^2 = 1 < 25$
$2\in \text{N}\because {2}^2 = 4 < 25$
$3\in \text{N}\because {3}^2 = 9 < 25$
$4\in \text{N}\because {4}^2 = 16 < 25$
Hence, the above set can be written as {1, 2, 3, 4}.

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Question 601 Mark

The given following sets are finite & in which of it infinite in it?
Set of concentric circles in a plane.

Answer

Infinite, since with a common centre infinitely many circles can be drown in a plane.

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Question 611 Mark

Are the following sets equal?
A = {x : x is a letter in the word paper},
B = {x : x is a letter in the word paper},
C = {x : x is a letter in the word paper}.

Answer

A = {a, e, p, r}
B = {a, e, p, r} (repetition of 'p' is not allowed)
C = {a, o, , r}
as A = B $\not=$ C, $\therefore$ The sets are not equal.

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Question 621 Mark

The given set is the example of an empty set or not?
{$x : x^2- 2 = 0$ and $x$ is rational}

Answer

$\{\text{x}^2-2=0\Rightarrow\text{x}^2=2\Rightarrow\text{x}=\pm\sqrt{2}\in\text{Q}\},$ the set of rational numbers So, this set is empty.

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Question 631 Mark

The given collections are sets or not? Justify your answer:
The collection of most talented writers of India.

Answer

It is not a set as the term 'most' is not well defined. A writer may be talented in the eye of one person, but he may not be talented in the eye of some other person.

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Question 641 Mark

If $A=\left\{(x, y): y=e^x, x \in R\right\}$ and $B=\left\{(x, y): e^{-x}, x \in R\right\}$, then write $A \cap B$.

Answer

$\text{A}\cap\text{B}=\Big\{\text{(x, y) : y = e}^\text{x},\text{x}\in\text{R}\Big\}\cap\Big\{\text{(x, y) : y = e}^\text{-x},\text{x}\in\text{R}\Big\}$
$\Rightarrow\text{A}\cap\text{B}=\Big\{\text{(x, y) : y = e}^\text{x}\text{(x, y) : y = e}^\text{-x},\text{x}\in\text{R}\Big\}$
$\Rightarrow\text{A}\cap\text{B}=\Big\{(0, 1) :\text{y = 1 = e}^0=\text{e}^{-0},\text{x}= 0\in\text{R}\Big\}$
$\Rightarrow\text{A}\cap\text{B}=\Big\{(0, 1)\Big\}.$

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Question 651 Mark

Describe the following sets in Roster form:
{x ∈ N : x is a prime number, 10 < x < 20};

Answer

In Roster form, we describe a set by listing its elements, separated by commas and the elements are written within braces { }. If a set has infinitely many elements, them comma is followed by ..., where the dots stand for 'and so on'. We note that a < x < b. The prime numbers which are more than 10 fact less than 20 are 11, 13, 17 and 19.
Hence the above set can be written as {11, 13, 17, 191}

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Question 661 Mark

The given set is the example of an empty set or not?
{x : x is a point common to any two parallel lines}.

Answer

This set is empty as any two parallel lines never intersect each other.

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Question 671 Mark

If A and B are two sets such that $\text{A}\subset\text{B},$ then find:$\text{A}\cup\text{B}$

Answer

$\text{A}\cup\text{B}$ denotes intersection of the two sets A and B, which consists of elements which are common to both A and B.
Since $\text{A}\subset\text{B,}$ every dement of A is already an element of B.
$\therefore\text{A}\cap\text{B}=\text{B}.$

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Question 681 Mark

The given collections are sets or not? Justify your answer:
A collection of novels written by Munshi, Prem, Chand.

Answer

It forms a set as it is well defined.

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Question 691 Mark

How many elements has P(A), if $\text{A}=\phi$?

Answer

$\because$ an empty set has zero element.
$\therefore$ power set of $\phi$ has $2^0 = 1$ element.

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Question 701 Mark

The given following sets are finite & in which of it infinite in if?
$\{\text{x}\in\text{N}:\text{x}<200\}$

Answer

Finite, $\because\{\text{x}\in\text{N}:\text{x}<200\}$ = {1, 2, 3,...199} which is finite.

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Question 711 Mark

List all the element of the following sets:
$\text{C}=\Big\{\text{x : x is integer,}-\frac{1}{2}<\text{x}<\frac{1}{9}\Big\};$

Answer

The integers which lie between $\frac{-1}{2}$ and $\frac{9}{2}$ are 0, 1, 2, 3, 4
Hence C = {0, 1, 3, 4}.

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Question 721 Mark

The given following sets are finite & in which of it infinite in if?
Set of letters of the English Alphabets;

Answer

Finite, as there are only 26 letters of English alphabet.

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Question 731 Mark

The given following sets are finite & in which of it infinite in if?
$\{\text{x}\in\text{N}:\text{x}<5\}$

Answer

Infinite.
$\because\{\text{x}\in\text{N}:\text{x}<5\}$ = {...,-3, -2, -1, 0, 1, 2, 3, 4} which is infinite.

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Question 741 Mark

Write down the all possible proper subset of the given set.
{1, 2}

Answer

We know that if A is a set and B a subset of A, then B is called a proper subset of A. If $\text{B}\subseteq\text{A}$ and $\text{B}\not=\text{A},\phi$ and is written as $\text{B}\subset\text{A or B}\subseteq\text{A}.$
Hence, the proper subset are given by {1}, {2}.

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Question 751 Mark

Let A and B be two sets having 4 and 7 elements respectively. Then write the maximum number of element that $\text{A}\cup\text{B}$ can have.

Answer

$\text{A}\cup\text{B}=\{\text{x : x}\in\text{A or x}\in\text{B}\}$
$\text{If A}\subset\text{B then A}\cup\text{B = \{x : x}\in\text{B\}}$
$\text{n(A}\cup\text{B)}=7$
$\text{If A}\cap\text{B}\not=\{\}$
$\text{then n(A}\cup\text{B)}<11$
$\text{If A}\cap\text{B}=\{\}$
$\text{then n (A}\cup\text{B)}=11$
The maximum number of elements that $\text{A}\cup\text{B}$ can have is 11.

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Question 761 Mark

Let A and B be two sets having 3 and 6 elements respectively. Write the minimum number of elements that $\text{A}\cup\text{B}$ can have.

Answer

$\text{A}\cup\text{B}=\{\text{x : x}\in\text{A or x}\in\text{B}\}$
$\text{If A}\subset\text{B then A}\cup\text{B}=\{\text{x : x}\in\text{A}\}$
$\text{n(A}\cup\text{B)}=6$
$\text{If A}\not\subset\text{B = \{x : x}\in\text{A or x}\in\text{B}\}$
$\text{n(A}\cup\text{B)}>6$
The minimum number of elements that $\text{A}\cup\text{B}$ can have is 6.

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Question 771 Mark

Describe the following sets in Roster form:
{x : x is a prime number which is a divisor of 60}

Answer

In Roster form, we describe a set by listing its elements, separated by commas and the elements are written within braces { }. If a set has infinitely many elements, them comma is followed by ..., where the dots stand for 'and so on'. The Prime divisors of 60 are 2, 3, 5.
Hence the above set can be written as {2, 3, 5}.

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Question 781 Mark

Prove thate: $\text{A}\subseteq\text{B},\ \text{B}\subseteq\text{C}$ and $\text{C}\subseteq\text{A}\Rightarrow\text{A}\subseteq\text{C}.$

Answer

We have $\text{A}\subseteq\text{B},\ \text{B}\subseteq\text{C}$ and $\text{C}\subseteq\text{A},$ so $\text{A}\subseteq\text{B}\subseteq \text{C}\subseteq\text{A}$
Now, A is a subset of B and B is a subset of C, so A is a subset of C, i.e., $\text{A}\subseteq\text{C}$
Also, $\text{C}\subseteq\text{A}$
Hence, $\text{A}\subseteq\text{C}.$

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Question 791 Mark

List all the element of the following sets:
F = {x : x is a letter of the word "MISSISSIPPI"};

Answer

The distinct letters of the word 'MISSISSIPPI' are M, I, S, P
Hence F = {M, I, S, P}.

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Question 801 Mark

If $\text{X}=\{8^\text{n}-7\text{x}-1:\text{n}\in\text{N}\}$ and $\text{Y}=\{49(\text{n}-1):\text{n}\in\text{N}\},$ then prove thet $\text{X}\subseteq\text{Y}.$

Answer

$\text{X}=\{8^\text{n}-7\text{x}-1:\text{n}\in\text{N}\}$ $\text{Y}=\{49(\text{n}-1):\text{n}\in\text{N}\}$ In order to show that $\text{X}\subseteq\text{Y}$ we shoe the every element of X is an element of Y. So let $\text{x}\in\text{X}\Rightarrow\text{x}=8^\text{n}-7\text{m}-1$ for same $\text{m}\in\text{N}$ $\Rightarrow\text{x = (1 + 7)}^\text{m}- 7\text{m} - 1$ $=(^\text{m}\text{C}_01^\text{m}+^\text{m}\text{C}_11^\text{m-1}7+...+^\text{m}\text{C}_\text{m-1}1^17^\text{m-1}+^\text{m}\text{C}_\text{m}7^\text{m})-7\text{m}-1$ [using binomail expansion] $=1+7\text{m}+^\text{m}\text{C}_27^2+^\text{m}\text{C}_37^3+...+^\text{m}\text{C}_\text{m}7^\text{m}-7\text{m}-1$ $=\ ^\text{m}\text{C}_27^2+\ ^\text{m}\text{C}_37^3+...+\ ^\text{m}\text{C}_\text{m}7^\text{m}$ $=49(^\text{m}\text{C}_2+^\text{m}\text{C}_3+...+^\text{m}\text{C}_\text{m}7^\text{m}),\ \text{m}\geq2 $ $=49\text{t}_\text{m},\ \text{m}\geq2,$ where $\text{t}_\text{m}=^\text{m}\text{C}_2+^\text{m}\text{C}_37+...+^\text{m}\text{C}_\text{m}7^\text{m-2}$ Is some positive integer depending on $\text{m}\geq2$ For $\text{m} = 1$$\text{x} = 1^8 - 7 × 1 - 1$
$=8 - 8$
$= 0$
Hence, X contains all positive integral multiples of 49. Also, Y consistes all positive integral multiples of 49, including 0, for n = 1. Thuse, we coclude that $\text{X}\subseteq\text{Y}.$

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Maths Solve the following Question.(1 Marks)