Question 12 Marks
List all the elements of the following sets:
$\text{A} = \{\text{x : x 2} \leq 10,\text{ x} \in\text{ Z}\};$
AnswerThe entegers whose squares are less then or equal to 10 are:
(-3)2 = 9 < 10
(-2)2 = 4 < 10
(-1)2 = 1 < 10
02 = 0 < 10
12 = 1 < 10
22 = 4 < 10
32 = 3 < 10
The square of other integers are more than 10.
Hence $\text{A}=\{0, \pm1,\pm2,\pm3\}$
OR
A = {0, -1, -2, -3, 1, 2, 3}.
View full question & answer→Question 22 Marks
List all the element of the following sets:
E = {x : x is a month of a year not having 31 days};
AnswerA month has either 28, 29, 30 or 31 day.
Out of the 12 months in a year, the months that have 31 days are:
January, March, May, July, August, October, Desember.
$\therefore$ E = {Fabruary, April, June, September, November}.
View full question & answer→Question 32 Marks
List all the element of the following sets:
D = {x : x is a vowel in the word "EQUATION"};
AnswerThe vowel in the word EQUATION are E, U, I, O, A.
Since the order in which the elements of a set are written is unmaterial, D = {E, U, I, O, A}.
View full question & answer→Question 42 Marks
Let A = {x : x $\in$ N}, B = {x : x = 2n, n $\in$ N}, C = {x : x = 2n - 1, n $\in$ N} and D = {x : x is a prime natural number}. Find:
$\text{A}\cap\text{D}$
AnswerWe have, A = {x : x $\in$ N} = {1, 2, 3,.....}, the set of natural numbers D = {x : x is a prime natural number}
= {2, 3, 5, 7,....}, the set of odd natural numbers $\therefore\text{A}\cap\text{D}=\{\text{x : x}\in\text{A and x}\in\text{D}\}$ $=\text{D }[\because\text{D}\subset\text{A}].$ View full question & answer→Question 52 Marks
Let A = {x : x $\in$ N}, B = {x : x = 2n, n $\in$ N}, C = {x : x = 2n - 1, n $\in$ N} and D = {x : x is a prime natural number}. Find:
$\text{A}\cap\text{C}$
AnswerWe have,
A = {x : x $\in$ N}
= {1, 2, 3,.....}, the set of natural numbers
C = {x : x = 2n - 1, n $\in$ N}
= {1, 3, 5,....}, the set of odd natural numbers
$\therefore\text{A}\cap\text{C}=\{\text{x : x}\in\text{A and x}\in\text{C}\}$
= C $[\therefore\text{C}\subset\text{A}].$
View full question & answer→Question 62 Marks
Let A = {1, 2, 4, 5}, B = {2, 3, 5, 6}, C = {4, 5, 6, 7}. Verify the following identities:
$\text{A}\cup(\text{B}\cap\text{C})=(\text{A}\cup\text{B})\cap(\text{A}\cup\text{C})$
Answer$\text{A} = \{1, 2, 4, 5\}, $
$\text{B} = \{2, 3, 5, 6\},$
and $\text{C}= \{4, 5, 6, 7\}$
$\text{B}\cap\text{C}= \{5, 6\}$
$\text{A}\cup(\text{B}\cap\text{C})= \{1, 2, 4, 5, 6\} ....(1)$
$(\text{A}\cup\text{B})= \{1, 2, 3, 4, 5, 6\}$
$(\text{A}\cup\text{C})= \{1, 2, 4, 5, 6, 7\}$
$(\text{A}\cup\text{B})\cap(\text{A}\cup\text{C})=\{1, 2, 4, 5, 6\} ....(2)$
From eqn (1) and eqn (2), we get
$\text{A}\cup(\text{B}\cap\text{C})=(\text{A}\cup\text{B})\cap(\text{A}\cup\text{C}).$
View full question & answer→Question 72 Marks
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:
$\text{B}\cup\text{C}\cup\text{D}$
Answer$\text{B}\cup\text{C}\cup\text{D}=\{\text{x | x}\in\text{B or x}\in\text{C or x}\in\text{D}\}$
= {4, 5, 7, 8, 9, 10, 11, 12, 13, 14}.
View full question & answer→Question 82 Marks
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:
$\text{A}\cup\text{B}\cup\text{D}$
Answer$\text{A}\cup\text{B}\cup\text{D}=\{\text{x | x}\in\text{A or x}\in\text{B or x}\in\text{D}\}$
= {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14}.
View full question & answer→Question 92 Marks
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:
$\text{A}\cup\text{B}\cup\text{C}$
Answer$\text{A}\cup\text{B}\cup\text{C}=\{\text{x | x}\in\text{A or x}\in\text{B or x}\in\text{C}\}$
= {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}.
View full question & answer→Question 102 Marks
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:
$\text{A}\cap(\text{B}\cup\text{C})$
Answer$\text{A}\cap(\text{B}\cup\text{C})$ = all those elements which are common to A and $\text{B}\cup\text{C}=\{\text{x | x}\in \text{A and x}\in\text{B}\cup\text{C}\}$
Now, $\text{B}\cup\text{C}$ = {4, 5, 6, 7, 8, 9, 10, 11}
$\therefore\text{A}\cap(\text{B}\cup\text{C})=\{1, 2, 3, 4, 5\}\cap\{4, 5, 6, 7, 8, 9, 10, 11\}$
= {4, 5}.
View full question & answer→Question 112 Marks
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:
$(\text{A}\cap\text{B})\cap(\text{B}\cap\text{C})$
Answer$(\text{A}\cap\text{B})\cap(\text{B}\cap\text{C})=\{\text{x | x}\in(\text{A}\cap\text{B) and x }\in (\text{B}\cap\text{C})\}$
Now, $\text{A}\cap\text{B}=\{\text{x | x}\in \text{A and x}\in\text{B}\}$
i.e., elements which are common to A & B
$\therefore \text{A}\cap\text{B}=\{1, 2, 3, 4, 5\}\cap\{4, 5, 6, 7, 8\}$
= {4, 5}
Also
$\therefore \text{B}\cap\text{C}=\{4, 5, 6, 7, 8\}\cap\{7, 8, 9, 10, 11\}$
= {7, 8}
Hence, $(\text{A}\cap\text{B})\cap(\text{B}\cap\text{C})=\{4, 5\}\cap\{7,8\}$
= $\oint$ [$\therefore$ there is no element common in {4, 5} and {7, 8}].
View full question & answer→Question 122 Marks
For any two sets A and B, show that the following statements are equevalent:
$\text{A}\cup\text{B}=\text{B}.$
AnswerWe new show that (3) ⇒ (4)
So assume that $\text{A }\cup\text{B}=\text{B}$
To show: $\text{A}\cap\text{B}=\text{A}$
$\because\text{A}\cup\text{B}=\text{B}$
$\text{A}\subset\text{B}$ and so $\text{A}\cap\text{B}=\text{A}$
So (3) ⇒ (4) is true.
View full question & answer→Question 132 Marks
For any two sets A and B, prove that.
$\text{B}\subset\text{A}\cup\text{B}.$
AnswerLet $\text{x}\in\text{B}.$ Then
$\Rightarrow\text{x}\in\text{B}\cup\text{A}$
$\Rightarrow\text{x}\in\text{A}\cup\text{B}$
$\therefore\text{B}\subset(\text{A}\cup\text{B}).$
View full question & answer→Question 142 Marks
For any two sets A and B, prove that.
$\text{A}\subset\text{B}\Rightarrow\text{A}\cap\text{B}=\text{A}.$
AnswerLet $\text{x}\in\text{A}\subset\text{B}.$ Then
$\Rightarrow\text{x}\in\text{B}$
Let and $\text{x}\in\text{A}\cap\text{B}$
$\Leftrightarrow\text{x}\in\text{A and x}\in\text{B}$
$\Leftrightarrow\text{x}\in\text{A and x}\in\text{A}$ $(\because\text{A}\subset\text{B})$
$\therefore(\text{A}\cap\text{B})=\text{A}.$
View full question & answer→Question 152 Marks
Find the smallest set A such that $\text{A}\cup\{1, 2\} = \{1, 2, 3, 5, 9\}.$
AnswerThe smallest set A such that $\text{A}\cup\{1, 2\} = \{1, 2, 3, 5, 9\}$ is {3, 5, 9}
$\because\{3, 5, 9\}\cup\{1, 2\} = \{1, 2, 3, 5, 9\}$
Any other set B such that $\text{B}\cup\{1, 2\} = \{1, 2, 3, 5, 9\}$ will contain A.For example we contake B to be {1, 3, 5, 9} or {1, 2 3, 5, 9}.
Clearly B contains A = {3, 5, 9}.
View full question & answer→Question 162 Marks
Describe the following sets in set-builder form:
D = {10, 11, 12, 13, 14, 15};
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'.
$\text{D} = \{\text{x} \in\text{ N}: 9 <\text{ x} < 16\},$
i.e. D is the set of natural numbers which are more than 9 but less than 16.
View full question & answer→Question 172 Marks
Describe the following sets in set-builder form:
C = {0, 3, 6, 9, 12, ...};
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'.$\text{C} = \{\text{x : x = 3k, k} \in\text{ Z}^+,\text{ the set of positive integers}\},$
i.e. C is the set of multiples of 3 including 0.
View full question & answer→Question 182 Marks
Describe the following sets in set-builder form:
A = {1, 2, 3, 4, 5, 6}
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 $\because$ or 'I' is read as 'such that'. So the above set A in Set-Builder form may be written as $\text{A} = \{\text{x} \in\text{ N}:\text{x} < 7\}$ i .e A is the set of natural numbers x such that x is less than 7. Or
$\text{A} = \{\text{x} \in\text{N } | 1 < \text{x} < 6\},$ i.e. A is the set of natural numbers x such that x is greater than or equal 1 and less than or equal to 6. View full question & answer→Question 192 Marks
Describe the following sets in Roster form:
{x : x is a two digit number such that the sum of its digits is 8}
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 (17, 26, 35, 44, 53, 62, 71, 80}.
View full question & answer→Question 202 Marks
Write the sets $\Big\{\frac{1}{2},\frac{2}{5},\frac{3}{10},\frac{4}{17},\frac{5}{26},\frac{6}{37},\frac{7}{50}\Big\}$ in the set-builder form.
AnswerThe set-builder form of the set $\Big\{\frac{1}{2},\frac{2}{5},\frac{3}{10},\frac{4}{17},\frac{5}{26},\frac{6}{37},\frac{7}{50}\Big\}\text{ is }\Big\{\frac{\text{n}}{\text{n}^2+1}:\text{n}\in\text{N, n}\le7\Big\}.$
View full question & answer→