Sets, Relations and Functions Test

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Sets, Relations and Functions Test - 1

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Weekly Quiz Competition

Question 1
2 / -0.83

Let X = {x | x = 2 + 4k, where k = 0, 1, 2, 3,...24}. Let S be a subset of X such that the sum of no two elements of S is 100. What is the maximum possible number of elements in S ?

A
10

B
11

C
13

D
12

Solution

Calculation:
The set X is given by
{2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98}.
We want to find the maximum size of a subset S of X such that no two
elements sum to 100.
The pairs in X that sum to 100 are
(2, 98), (6, 94), (10, 90), (14, 86), (18, 82), (22, 78), (26, 74), (30, 70), (34,
66), (38, 62), (42, 58), (46, 54).
Therefore,
S = {2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50}
∴The maximum possible number of elements in S be 13.
Question 2
2 / -0.83

In a party of 150 persons, 75 persons take tea, 60 persons take coffee and 50 persons take milk. 15 of them take both tea and coffee, but no one taking milk takes tea. If each person in the party takes at least one drink, then what is the number of persons taking milk only ?

A
50

B
40

C
30

D
20

Solution

Let, x number of persons taking milk only.
According to the question
60 + 15 + (x - 5) + (50 - x) + x = 150
120 + x = 150
x = 150 - 120 = 30
∴The required value is 30.
Question 3
2 / -0.83

Which of the following is a set?

A
The collection of all the months of a year.

B
The collection of all interesting novels.

C
The collection of all rich persons of Delhi.

D
A collection of 10 best writers of India.

Solution

(a) The collection of all months of a year is a well-defined collection of objects because one can definitely identify a month that belongs to this collection. Hence, this collection is a set.
(b) A collection of novels is not a well-defined collection because one cannot identify a book that belongs to this collection. Hence, this collection is not a set.
(c) A collection of top rich persons can not be defined. Hence, this collection is not a set.
(d) The collection of the ten most talented writers of India is not a well-defined collection because the criteria for determining a writer 's talent may vary from person to person. Hence, this collection is not a set.
Question 4
2 / -0.83

If Q = {x : x = 1 / y, where y ∈ N}, then:

A
0 ∈Q

B
1 ∈Q

C
2 ∈Q

D
2 / 3 ∈Q

Solution

As y ∈N , y can be 1, 2, 3, 4...
∴x will be 1, 1/2, 1/3, 1/4...
⇒1 ∈Q
Question 5
2 / -0.83

Write A = {1, 4, 9, 16, 25} in set builder form.

A
A = {x : x is a prime number}

B
A = {x : x is the cube of a natural number}

C
A = {x : x is the square of a natural number}

D
A = {x : x is an even natural number}
Solution
We know that, 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25
Therefore the set A = {1, 4, 9, 16, 25...} can be written in set builder form as:
A = {x: x is the square of a natural number}
Question 6
2 / -0.83

A = { x : x ≠ x } represents:

A
{0}

B
{1}

C
{x}

D
{ }
Solution
According to this, x should be a number which is not equal to the number itself and there is no number which is not equal to the number itself.
Therefore A is an empty set .
Question 7
2 / -0.83

If A = {3, 6, 9, 12} and B = {6, 8, 9} then intersection of A and B is:

A
{3, 6}

B
{3, 12}

C
{6, 9}

D
{9, 12}
Solution
Anintersection is the collection of all the elements that are common to all the sets under consideration.
Here element6 &9 are common in both the sets. So option C is correct.
Question 8
2 / -0.83

Which of the following is a finite set?

A
Set of roots of the equation x2–25 = 0.

B
Set of natural numbers.

C
The set of lines parallel to y –axis.

D
The set of numbers which are multiples of 5.
Solution
A finite set is a set that has a finite number of elements.
Since x² –25 = 0 has finite number of elements. So it is a finite set.
Question 9
2 / -0.83

Consider the following statements:
I: If A = {x: x is an even natural number} and B = {y: y is a natural number}, A subset B.
II: Number of subsets for the given set A = {5, 6, 7, 8} is 12.
III: Number of proper subsets for the given set A = {5, 6, 7, 8} is 15.
Which of the following statement(s) is/are correct?

A
I and II

B
II and III

C
I and III

D
None of the above

Solution

Concept:
The null set is a subset of every set. (ϕ⊆A)
Every set is a subset of itself. (A ⊆A)
The number of subsets of a set with n elements is 2ⁿ .
The number of proper subsets of a given set is 2ⁿ - 1
Calculation:
Statement I: If A = {x: x is an even natural number} and B = {y: y is a natural number}, A subset B.
A = {2, 4, 6, 8, 10, 12, ...} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...}.
It is clear that all the elements of set A are included in set B.
So, set A is the subset of set B.
Statement I is correct.
Statement II: Number of subsets for the given set A = {5, 6, 7, 8} is 16.
Given: A = {5, 6, 7, 8}
The number of elements in the set is 4
We know that,
The formula to calculate the number of subsets of a given set is 2ⁿ
= 2⁴ = 16
Number of subsets is 16
Statement II is incorrect.
Statement III: Number of proper subsets for the given set A = {5, 6, 7, 8} is 15.
The formula to calculate the number of proper subsets of a given set is 2ⁿ - 1
= 2⁴ - 1
= 16 - 1 = 15
The number of proper subsets is 15.
Statement III is correct.
∴Statements I and III are correct.
Question 10
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Which of the following has only one subset?

A
{ }

B
{5}

C
{4, 5}

D
{0}
Solution
Every set has the empty set as a subset. So if a set has 1 element, like {0}, then it will have 2 subsets: itself and the empty set, which is denoted by{ } .
So, if a set has only one subset , then this set must be the empty set .

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Sets, Relations and Functions Test - 1

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Sets, Relations and Functions Test - 1

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SHARING IS CARING

Question 1

10

11

13

12

Solution

Question 2

50

40

30

20

Solution

Question 3

The collection of all the months of a year.

The collection of all interesting novels.

The collection of all rich persons of Delhi.

A collection of 10 best writers of India.

Solution

Question 4

0 ∈Q

1 ∈Q

2 ∈Q

2 / 3 ∈Q

Solution

Question 5

A = {x : x is a prime number}

A = {x : x is the cube of a natural number}

A = {x : x is the square of a natural number}

A = {x : x is an even natural number}

Solution

Question 6

{0}

{1}

{x}

{ }

Solution

Question 7

{3, 6}

{3, 12}

{6, 9}

{9, 12}

Solution

Question 8

Set of roots of the equation x2–25 = 0.

Set of natural numbers.

The set of lines parallel to y –axis.

The set of numbers which are multiples of 5.

Solution

Question 9

I and II

II and III

I and III

None of the above

Solution

Question 10

{ }

{5}

{4, 5}

{0}

Solution

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