Question 511 Mark
Write the number of elements in tha power set of null set.
AnswerIf A is the void set $\phi$, then P(A) has just one element $\phi$ i.e. $\text{P}(\phi) =\{\phi\}.$
View full question & answer→Question 521 Mark
Write down the all possible subset of the given set.
$\{1,\{1\}\} .$
AnswerThe set has 2 element, so power set has $2^2=4$ elemente, namely, $\left.\phi,\{1\},\{11\}\right\},\{1,\{1\}\}$.
View full question & answer→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.
AnswerIt forms a set as it is well defined.
View full question & answer→Question 541 Mark
The given set is the example of an empty set or not?
Set of all even natural numbers divisible by 5.
AnswerThis set is non-empty as 10 is an even natural number divisible by 5.
View full question & answer→Question 551 Mark
From the sets given below, select equal sets and equivalent sets.
$A = \{0, a\},$
$B = \{1, 2, 3, 4\},$
$C = \{4, 8, 12\},$
$D = \{3, 1, 2, 4\},$
$E = \{1, 0\},$
$F = \{8, 4, 12\},$
$G = \{1, 5, 7, 11\},$
$H = \{a, b\}.$
AnswerEqual 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 $F$
|
$\{\because n(C) = n(F) = 3\}$.
|
View full question & answer→Question 561 Mark
The given following sets are finite & in which of it infinite in if?
$\{\text{x}\in\text{N}:\text{x}>5\}$
AnswerFinite, $\because\{\text{x}\in\text{N}:\text{x}>5\}$ = {6, 7, 8,...} which is infinite.
View full question & answer→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$\text{n(A}\cup\text{B)}=\text{n(A) + n(B) - n(A}\cap\text{B)}$
$\text{n(A}\cap\text{B)}=\text{n(A) + n(B) - n(A}\cup\text{B)}$
$=20+25-40$
$=5.$
View full question & answer→Question 581 Mark
The set of all positive integers whose cube is odd.
AnswerAs 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 $\{\text{x}\in \text{Z},\text{x = 2n+1},\text{n}\ge0\}.$
View full question & answer→Question 591 Mark
Describe the following sets in Roster form:
$\{\text{x}\in\text{N}:\text{x}^2 < 25\};$
AnswerIn 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}.
View full question & answer→Question 601 Mark
The given following sets are finite & in which of it infinite in it?
Set of concentric circles in a plane.
AnswerInfinite, since with a common centre infinitely many circles can be drown in a plane.
View full question & answer→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}.
AnswerA = {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.
View full question & answer→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.
View full question & answer→Question 631 Mark
The given collections are sets or not? Justify your answer:
The collection of most talented writers of India.
AnswerIt 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.
View full question & answer→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$\text{A}\cap\text{B}=\Big\{\text{(x, y) : y = e}^\text{x},\text{x}\in\text{R}\Big\}\cap\Big\{\text{(x, y) : y = e}^\text{-x},\text{x}\in\text{R}\Big\}$
$\Rightarrow\text{A}\cap\text{B}=\Big\{\text{(x, y) : y = e}^\text{x}\text{(x, y) : y = e}^\text{-x},\text{x}\in\text{R}\Big\}$
$\Rightarrow\text{A}\cap\text{B}=\Big\{(0, 1) :\text{y = 1 = e}^0=\text{e}^{-0},\text{x}= 0\in\text{R}\Big\}$
$\Rightarrow\text{A}\cap\text{B}=\Big\{(0, 1)\Big\}.$
View full question & answer→Question 651 Mark
Describe the following sets in Roster form:
{x ∈ N : x is a prime number, 10 < x < 20};
AnswerIn 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}
View full question & answer→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}.
AnswerThis set is empty as any two parallel lines never intersect each other.
View full question & answer→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.
$\therefore\text{A}\cap\text{B}=\text{B}.$
View full question & answer→Question 681 Mark
The given collections are sets or not? Justify your answer:
A collection of novels written by Munshi, Prem, Chand.
AnswerIt forms a set as it is well defined.
View full question & answer→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.
View full question & answer→Question 701 Mark
The given following sets are finite & in which of it infinite in if?
$\{\text{x}\in\text{N}:\text{x}<200\}$
AnswerFinite, $\because\{\text{x}\in\text{N}:\text{x}<200\}$ = {1, 2, 3,...199} which is finite.
View full question & answer→Question 711 Mark
List all the element of the following sets:
$\text{C}=\Big\{\text{x : x is integer,}-\frac{1}{2}<\text{x}<\frac{1}{9}\Big\};$
AnswerThe integers which lie between $\frac{-1}{2}$ and $\frac{9}{2}$ are 0, 1, 2, 3, 4
Hence C = {0, 1, 3, 4}.
View full question & answer→Question 721 Mark
The given following sets are finite & in which of it infinite in if?
Set of letters of the English Alphabets;
AnswerFinite, as there are only 26 letters of English alphabet.
View full question & answer→Question 731 Mark
The given following sets are finite & in which of it infinite in if?
$\{\text{x}\in\text{N}:\text{x}<5\}$
AnswerInfinite.
$\because\{\text{x}\in\text{N}:\text{x}<5\}$ = {...,-3, -2, -1, 0, 1, 2, 3, 4} which is infinite.
View full question & answer→Question 741 Mark
Write down the all possible proper subset of the given set.
{1, 2}
AnswerWe 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}.
View full question & answer→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.
View full question & answer→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.
View full question & answer→Question 771 Mark
Describe the following sets in Roster form:
{x : x is a prime number which is a divisor of 60}
AnswerIn 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}.
View full question & answer→Question 781 Mark
Prove thate: $\text{A}\subseteq\text{B},\ \text{B}\subseteq\text{C}$ and $\text{C}\subseteq\text{A}\Rightarrow\text{A}\subseteq\text{C}.$
AnswerWe 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, $\text{A}\subseteq\text{C}.$
View full question & answer→Question 791 Mark
List all the element of the following sets:
F = {x : x is a letter of the word "MISSISSIPPI"};
AnswerThe distinct letters of the word 'MISSISSIPPI' are M, I, S, P
Hence F = {M, I, S, P}.
View full question & answer→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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