Question 11 Mark
Show that the set of letters needed to spell "CATARACT" and the set of letters needed to spell "TRACT" are equal?
AnswerLetters required to spell CATARACT are {C, A, T, R}. Let this set be denoted as E. E = {C, A, T, R} Letters required to spell TRACT are {T, R, A, C}. Let this set be denoted as F. F = {T, R, A, C} The two sets E & F are equal because every element of E is a member of F & every element of F is a member of E.
View full question & answer→Question 21 Mark
Write down the all possible subset of the given set. {a, b, c}
AnswerThe set has 3 elements, so power set has $2^3 = 8$ alements, namely $\phi,$ {a}, {b}, {c}, {a ,b}, {b, c}, {a, c}, {a, b, c}.
View full question & answer→Question 31 Mark
Describe the following sets in set-builder form: {5, 25, 125 625};
AnswerIn set builder form, a set is describerd by some characterising property p(x) of its elements x. In this case a set can be described as {x : p((x) hold}. OR {x | p(x) hold} which are read as 'The set of all x such that p(x) hold'. The symbols ':' or 'I' is read as 'such that'. $\because {5}^1=5\\{5}^2=25\\{5}^3=125\\{5}^4=625$ $\therefore$ The above se can be describded as $\{\text{x : x}={5}^\text{n},1\leq\text{n}\leq4\}.$
View full question & answer→Question 41 Mark
If $A=\left\{x \in C: x^2=1\right\}$ and $B=\left\{x \in C: x^4=1\right\}$, then write $A-B$ and $B-A$.
Answer$\text{A - B =\{x}\in\text{: x}\not\in\text{B}\}$ $=\Big\{\text{x}\in\text{C : x}^2=1\text{ and x}\not=1\Big\}$ $=\phi$ $\text{B - A}=\Big\{\text{x}\in\text{B : x}\not\in\text{A}\Big\}$ $=\Big\{\text{x}\not\in\text{C : x}^4= 1\text{ and x}^2\not=1\Big\}$ $=\{\text{i, -i}\}.$
View full question & answer→Question 51 Mark
Write down the all possible subset of the given set. {a}.
AnswerWe know that, if a set has n element, then its power set has $2^n$ elements. Here, $n =1$, so there $2^1=2$ subset of given set. The possible subset are $\phi$, {a}.
View full question & answer→Question 61 Mark
Describe the following sets in set-builder form: E = {0};
AnswerIn set Builder form, a set is described by some characterizing property P (x) of its elements x. In this case a set can be described as : {x: P(x) hold} which is read as 'the set of all x such that P (x) holds'. The symbols ':' or 'I' is read as 'such that'. $\text{E} =\{\text{x} \in\text{ Z} : -1 < \text{x} < 1\}$OR
$\text{E} = \{\text{x} \in\text{ Z}: \text{x} = 0\}.$
View full question & answer→Question 71 Mark
Describe the following sets in Roster form: $\{\text{x}\in\text {R} :\text{5 x > x}\}$
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 know that given any $\text{x}\in\text{R,}$ x is always less than or equal to itself, i.e. $\text{x}\leq\text{x}$ Hence the above set is empty, i.e. $\phi.$
View full question & answer→Question 81 Mark
If A and B are two sets such that $\text{A}\subset\text{B},$ then find:$\text{A}\cap\text{B}$
Answer$\text{A}\cap\text{B}$ 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. $\therefore\text{A}\cup\text{B}=\text{A}.$
View full question & answer→Question 91 Mark
The given collections are sets or not? Justify your answer: The collection of all girls in your class.
AnswerIt forms a set as it is well defined.
View full question & answer→Question 101 Mark
The given collections are sets or not? Justify your answer: A collection of most dangerous animals of the world.
AnswerIt is not a set as the term 'most dangerous' is not well defined. The notion of dangerous animals differs from person to person.
View full question & answer→Question 111 Mark
The given collections are sets or not? Justify your answer: The collection of all question in this chapter.
AnswerIt forms a set as it is well defined.
View full question & answer→Question 121 Mark
If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11}, and D = {10, 11, 12, 13, 14}. Find: $\text{B}\cup\text{D}$
Answer$\text{B}\cup\text{D}=\{\text{x : x}\in\text{B x or}\in\text{D}\}$ = {4, 5, 7, 8, 9, 10, 11, 12, 13, 14}.
View full question & answer→Question 131 Mark
If A is any set, prove that: $\text{A}\subseteq\phi\Leftrightarrow\text{A}=\phi.$
AnswerThe symbol '⇔' stands for if and only if (in short if). In order to show that 2 sets A and B are equel we show htat $\text{A}\subseteq\text{B}$ and $\text{B}\subseteq\text{A}.$ We have $\text{A}\subseteq\phi.\ \because \phi$ is a subset of every set $\therefore\ \phi\subset\text{A}$ Hence $\text{A} =\phi$ To show the backward implication, suppose that $\text{A}= \phi$ $\because$ Every set is a subset of itself. $\therefore\ \phi=\text{A}\subseteq \phi$ Hence, proved.
View full question & answer→Question 141 Mark
Are the following pairs of sets equal? Give reasons. A = {x : x is a letter of the word " WOLF"}; B = {x : x is a letter of the word " FOLLOW"}.
AnswerA = {W, O, L, F} B = {F, O, L, W} Here, A = B because every element of A is a member of B & every element of B is a member of A.
View full question & answer→Question 151 Mark
Write down the all possible proper subset of the given set. {1, 2, 3}
AnswerThe proper subsets are given by {1}, {2}, {3}, {1, 2}{2, 3}{1, 3}.
View full question & answer→Question 161 Mark
List all the elements of the following sets: $\text{B}=\Big\{\text{x : x}=\frac{1}{\text{2n-1}},1\leq\text{n}\le5\Big\};$
AnswerLet's find the values of x $=\frac{1}{\text{2n-1}},\text{for }1\leq \text{n}\leq5 $ for n = 1, $\text{x}=\frac{1}{1}=1$ for n = 2, $\text{x}=\frac{1}{2\times2-1}=\frac{1}{4-1}=\frac{1}{3}$ for n = 3, $\text{x}=\frac{1}{2\times3-1}=\frac{1}{6-1}=\frac{1}{5}$ for n = 4, $\text{x}=\frac{1}{2\times4-1}=\frac{1}{8-1}=\frac{1}{7}$ for n = 5, $\text{x}=\frac{1}{2\times5-1}=\frac{1}{10-1}=\frac{1}{9}$ Hence, $\text{B}=\Big\{1,\frac{1}{3},\frac{1}{5},\frac{1}{7},\frac{1}{9}\Big\}.$
View full question & answer→Question 171 Mark
Are the following pairs of sets equal? Give reasons. $A=\{2,3\}, B=\left\{x: x\right.$ is a solution of $\left.x^2+5 x+6=0\right\}$;
AnswerA = {2, 3} B = {- 2, - 3} A is not equal to B because every element of A is not a member of B & every element of B is not a member of A.
View full question & answer→Question 181 Mark
The given collections are sets or not? Justify your answer: A collection of all natural numbers less than 50.
AnswerThe collection of all natural numbers less than 50 forms a set as it is well defined.
View full question & answer→Question 191 Mark
Write down the all possible subset of the given set. $\{\phi\}.$
AnswerThe set has 1 element, so power set has $^1$ = 2 elements, namely $\phi,\{\phi\}.$
View full question & answer→Question 201 Mark
What is the difference between a collection and a set? Give reasons to support your answer?
AnswerEach set is a collection, but each collection need not be a set, For example, a collection of beautiful women in Delhi is just a collection and not a set, for the term beautiful is not well defined. Only well defined collection of objects forms a set.
View full question & answer→Question 211 Mark
Describe the following sets in Roster form: The set of all letters in the word 'Better'.
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 distinct letters are B, E, T, R. Hence the set can be written as{B, E, T, R}.
View full question & answer→Question 221 Mark
If a set contains n elements, then write the number of elements in its power set.
AnswerLet A be a set. Then collection or family of all subsets of A is called the power set of A and is denoted by P(A). A set having n elements has $2^n$ subsets. Therefore, if A is a finite set having n elements, then P(A) has $2^n$ elements.
View full question & answer→Question 231 Mark
The given collections are sets or not? Justify your answer: The collection of difficult topics in mathematics.
AnswerIt is not a set as the term 'difficult' is not well defined. A topic may be difficult for one person but may not be difficult for another person, so the term 'difficult' is vague.
View full question & answer→Question 241 Mark
What is the total number of proper subset of a set consisting of n elements?
AnswerWe know that , if A is a set having n elements then power set of A. Namely P(A) has $2^n$ elements. Out of this A is not proper subset. Hence, the total number of proper subsets of a set consisting of n elements $2^n - 1$.
View full question & answer→Question 251 Mark
Describe the following sets in Roster form: The set of all letters in the word 'Trigonometry'.
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'. As repetition is not allowed in a set, the distinct letters are T, R, I, G, O, N, M, E, Y. Hence the above set can be written as {T, R, I, G, O, N, M, E, Y}
View full question & answer→Question 261 Mark
If A and B are two sets such that $\text{A}\subset\text{B},$ then write B' - A' in term of A and B.
AnswerLet, $\text{x}\in\text{B}'-\text{A}'$ $\Rightarrow\text{x}\in\text{B}'\text{and x}\not\in\text{A}'$ $\Rightarrow\text{x}\notin\text{B }\text{and x}\in\text{A}$ $\Rightarrow\text{x}\not\in\text{B }\text{and x}\in\text{B}$ $(\because\text{A}\subset\text{B)}$ So B' - A' is an empty set.
View full question & answer→Question 271 Mark
Which of the following sets are equal?
i. $A=\{1,2,3\}$
ii. $B=\left\{x \in R: x^2-2 x+1=0\right\}$
iii. $C=\{1,2,2,3\}$
iv. $D=\left\{x \in R: x^3-6 x^2+11 x-6=0\right\}$.
Answer$A=\{1,2,3\} B=\left\{x \in R:(x-1)^2=0\right\}=\{x \in R: x=1,1\}=\{1\} C=\{1,2,3\}$
$(\because$ repetition is not allowed in a set) $D$ $=\left\{x \in R: x^3-6 x^2+11 x-6=0\right\}=\left\{x \in R:(x-1)\left(x^2-5 x+6\right)=0\right\}[\because x=1$ satisfies the above equetion $]=\{x \in R:(x-$
1) $(x-2)(x-3)=0\}=\{x \in R: x=1,2,3\}=\{1,2,3\}$
Hence the set $A, C$ and $D$ are equal.
View full question & answer→Question 281 Mark
Write down the all possible subset of the given set. {0, 1}
AnswerThe set has 2 elements, so power set has $2^2 = 4$ elements, namely {0}, {1}, {0, 1}.
View full question & answer→Question 291 Mark
Which of the following sets are equal? A = {x : x $\in$ N, x, < 3}, B = {1, 2}, C = {3, 1}, D = {x : x $\in$ N, x is odd, x < 5}, E = {1, 2, 1, 1} F = {1, 1, 3}.
AnswerA = {1, 2} B = {1, 2} C = {3, 1} D = {1, 3} E = {1, 2, 1, 1} = {1, 2} F = {1, 1, 3} = {1, 3} $\therefore$ A = B = E and C = D = F.
View full question & answer→Question 301 Mark
Write the set of all vowels in the English alphabet which precede q.
AnswerThe vowels which precede q, that is, come before q are a, e, i, o. Hence the set of vowels in the English alphabet which precede q are {a, e, i, o}.
View full question & answer→Question 311 Mark
Let A = {x : x $\in$ N, x is a multiple of 3} and B = {x : x $\in$ N and x is a multiple of 5}. Write $\text{A}\cap\text{B}.$
Answer$\text{A}\cap\text{B}=\{\text{x : x}\in\text{N, x}\in\text{A and x}\in \text{B}\}$ = {x : x $\in$ N, x is a multiple of 3 and x is a multiple of 5} = {x : x $\in$ N, x is a multiple of 15}.
View full question & answer→Question 321 Mark
If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11}, and D = {10, 11, 12, 13, 14}. Find: $\text{B}\cup\text{C}$
Answer$\text{B}\cup\text{C}=\{\text{x : x}\in\text{A x or}\in\text{C}\}$ = {4, 5, 7, 8, 9, 10, 11}.
View full question & answer→Question 331 Mark
Describe the following sets in Roster form: {1, 4, 9, 16, ..., 100};
AnswerIn set Builder form, a set s described by some characterising property $P ( x )$ of its elements x . In this case a set can be described as $\{x$ : $P(x)$ hold $\}$ which is read as 'the set of all $x$ such that $P(x)$ holds'. The symbols ' $:$ ' or ' $l$ ' is read as 'such that'. As $1^2=12^2=43^2=9:: 10^2=100 \therefore$ The above set may be described as $\left\{ x ^2: x \in N 8 c 1 \leq x \leq 10\right\}$.
View full question & answer→Question 341 Mark
If A = {(x, y): y = $\frac{1}{\text{x}},$ 0 $\not=$ x $\in$ R} and B = {(x, y) = -x, x $\in$ R}, then write $\text{A}\cap\text{B.}$
AnswerLet, $\text{z}\in\text{A}\cap\text{B}$ $\Rightarrow\text{z}\in\text{A and}\text{ z}\in\text{B}$ $\Rightarrow\text{z}\in\Big\{\text{(x, y) : y =}\frac{1}{\text{x}},0\not=\text{x}\in\text{R}\Big\}$ and $\text{z}\in\Big\{\text{(x, y) : y}={\text{-x}},\text{x}\in\text{R}\Big\}$ $\Rightarrow\text{z}\in\Big\{\text{(x, y) : -x =}\frac{1}{\text{x}},0\not=\text{x}\in\text{R}\Big\}$ $\Rightarrow\text{z}\in\Big\{\text{(x, y)}:\text{x}^2 =-1,\text{ x}\in\text{R}\Big\}$ $\Rightarrow\text{z}\in\{\}$ $\therefore\text{A}\cap\text{B}$ is an empty set.
View full question & answer→Question 351 Mark
Describe the following sets in Roster form: {2, 4, 6, 8 .....};
AnswerIn set B builder from, a set of described by some characterising property p(x) of it elements x. In this case a set can be describ as {x : p(x) hold}. OR {x | p(x) hold} which is read as 'The set of all x such that p(x) hold'. The symbol ':' or 'I' is read as 'such that'. The given set can be described as $\{\text{x : x = 2n, n}\in \text{N}\}(\therefore 2, 4, 6,...\text{are multiples of 2}).$
View full question & answer→Question 361 Mark
What universal sets would you propose for the following:The set of isosceles triangles.
AnswerThe set of isosceles triangles forms a subset of the set of all triangles in the plane.
Hence the set of all triangles in the plane forms a universal set for the set of isosceles triangles.
View full question & answer→Question 371 Mark
The given following sets are finite & in which of it infinite in if? $\{\text{x}\in\text{R}:0<\text{x}<1\}$
Answer$\{\text{x}\in\text{R}:0<\text{x}<1\}$ is an infinite set. $\because$ an interwal is an infinite set.
View full question & answer→Question 381 Mark
Describe the following sets in Roster form: $\{\text{x} \in\text{N} :\text{x}=\text{2n, n}\in\text{N}\};$
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 above set can be written as {2, 4, 6, 8....} since all those natural numbers, which can be written as a multiple of 2 are the even natural numbers.
View full question & answer→Question 391 Mark
The given collections are sets or not? Justify your answer: The collection of prime integers.
AnswerIt forms a set as it is well defined
View full question & answer→Question 401 Mark
Write down the all possible proper subset of the given set. {1}.
AnswerThe only subset of the given set are $\phi\ \&\ \{1\}.$Hence, there are no proper subsets.
View full question & answer→Question 411 Mark
If A and B are two sets such that n(A) = 115, n(B) = 326 and n(A - B) = 47, then write $\text{n(A}\cup\text{B)}.$
Answer$\text{n(A}-\text{B)}=\text{n(A) - n(A}\cap\text{B)}$ $\text{n(A}\cap\text{B)}=\text{n(A) - n(A}-\text{B)}$ $\text{n(A}\cup\text{B)}=\text{n(A) + n(B) - n(A}\cap\text{B)}$ $\text{n(A}\cup\text{B)}=\text{n(A) + n(B)}-\Big[\text{n(A) - n(A}-\text{B)}\Big]$ $=\text{n(A) + n(B) - n(A) + n(A - B)}$ $=\text{n(B) + n(A - B)}$ $=326+47$ $=373.$
View full question & answer→Question 421 Mark
What universal sets would you propose for the following:The set of right triangles.
AnswerThe set of right triangles is a subset of the set of all triangles in the plane. So, the set of all triangles in the plane forms a universal set for the set of right triangles.
View full question & answer→Question 431 Mark
Describe the following sets in set-builder form: $\text{B}=\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, ...\};$
AnswerIn set Builder form, a set is described by some characterising property P(x) of its elements x. In this case a set can be described as {x : P(x) hold} which is read as 'the set of all x such that P (x) holds'. The symbols ':' or 'I' is read as 'such that'. $ \text{B}=\Big\{\text{x: x}=\frac{1}{\text{n}},\text{n}\in\text{N}\Big\}$ i.e. B is the set of all those x such that $\text{x}=\frac{1}{\text{n}}, \text{where}\text{ n}\in\text{N.}$
View full question & answer→Question 441 Mark
Describe the following sets in Roster form: {x : x is a letter before e in the English alphabet};
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 above set in Roster form can be written as {a, b, c, d}. Since the letters a, b, c, and d precedes e in the English alphabet.
View full question & answer→Question 451 Mark
From the sets given below, pair the equivalent sets: A = {1, 2, 3}, B = {t, p, q, r, s}, C = $\{\alpha,\beta,\gamma\},$ D = {a, e, i, o, u}.
AnswerTwo finite sets are said to be equivalent if they have the same number of elements. As A and C have same number of elements, and B and D also have same number of elements. $\therefore$ A is equivalent to C & 8 is equivalent to D.
View full question & answer→Question 461 Mark
The given collections are sets or not? Justify your answer: The collection of good hockey players in India.
AnswerIt is not a set as the term 'good' is not well defined.
View full question & answer→Question 471 Mark
The given set is the example of an empty set or not? {x : x is a natural number, x < 8 and simultaneously x > 12};
AnswerThis set is empty as there is no natural numbar x such that x < 8 and simultaneously x > 12.
View full question & answer→Question 481 Mark
If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11}, and D = {10, 11, 12, 13, 14}. Find: $\text{A}\cup\text{B}$
AnswerA = {1, 2, 3, 4, 5} B = {4, 5, 6, 7, 8} So, $\text{A}\cup\text{B}= \{\text{x : x}\in\text{A or x}\in \text{B}\}$ = {1, 2, 3, 4, 5, 6, 7, 8}.
View full question & answer→Question 491 Mark
The given set is the example of an empty set or not? Set of all even prime numbers.
AnswerAs 2 belongs to this set, so it is non-empty.
View full question & answer→Question 501 Mark
If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11}, and D = {10, 11, 12, 13, 14}. Find: $\text{A}\cup\text{C}$
Answer$\text{A}\cup\text{C}=\{\text{x : x}\in\text{A x}\in\text{C}\}$ = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}.
View full question & answer→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, $\phi,$ {1}, {{1}}, {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}
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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.
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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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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.
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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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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.
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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}.
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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.
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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.
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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}.
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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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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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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}.
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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, $\text{A}\subseteq\text{C}.$
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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}.
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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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