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All questions of Sets for JAMB Exam

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Let R be a non-empty relation on a collection of sets defined by ARB if and only if A  ∩ B = Ø
Then (pick the TRUE statement) 
  • a)
    R is relexive and transitive
  • b)
    R is symmetric and not transitive
  • c)
    R is an equivalence relation
  • d)
    R is not relexive and not symmetric
Correct answer is option 'B'. Can you explain this answer?

Vivek Patel answered
The correct answer is B 
Let, A={1,2,3}
B={4,5} 
C={1,6,7}

now, A∩B=∅ 
B∩C=∅ but  A∩C≠∅
Relation R is not transitive.
A∩A=A
R is not reflexive.
A∩B=B∩A
R is symmetric
So, 
A is false as R is not reflexive or transitive
B is true.
C is false because R is not transitive or reflexive
D is false because R is symmetric

 If R = ((1, 1), (3, 1), (2, 3), (4, 2)), then which of the following represents R2, where R2 is R composite R?
  • a)
    ((1, 1), (3, 1), (2, 3), (4, 2))
  • b)
    f(1, 1), (9, 1), (4, 9), (16, 4))
  • c)
    1(1, 3), (3, 3), (3, 4), (3, 2))
  • d)
    ((1, 1), (2, 1), (4, 3), (3, 1))
Correct answer is option 'D'. Can you explain this answer?

EduRev CAT answered
The correct answer is D as
 R = ((1, 1), (3, 1), (2, 3), (4, 2))
RoR=R2=((1, 1), (3, 1), (2, 3), (4, 2))((1, 1), (3, 1), (2, 3), (4, 2))
     =((1, 1), (3, 1), (2, 1), (4, 3))
take the first set (1,1) then take the second element of this subset check in the other set R is there any starting with 1 if yes then take its second element and make a subset in R2 similarly check for all.
like (4,2) (2,3)=(4,3)in R2

Number of subsets of a set of order three is
  • a)
    3
  • b)
    6
  • c)
    8
  • d)
    9
Correct answer is option 'C'. Can you explain this answer?

Lavanya Menon answered
A set with 'n' elements in it can have '2n' subsets.
eg: Let us consider a set A = {1,2,3}
The possible subsets are:
{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} and {}
Where, {} is the empty set.
Number of subset = 2n
order 3 = 23 ⇒ 8
Therefore, the number of subsets is 8.

Let R be a relation "(x -y) is divisible by m", where x, y, m are integers and m > 1, then R is
  • a)
    symmetric but not transitive
  • b)
    partial order
  • c)
    equivalence relation
  • d)
    anti symmetric and not transitive
Correct answer is option 'C'. Can you explain this answer?

Nikita Singh answered
a) Since x - x = 0, m
=> x - x is divisible by m
(x,x) ∈ R
=> R is reflexive
b) Let (x,y) ∈ R
=> x - y = mq for some q ∈ I
=> y - x = m(-q)
y - x is divisible by m
(y,x) ∈ R
=> R is symmetric.
c) Let (x,y) and (y,z) ∈ R
=> x - y is divisible by m and y - z is divisible by m
=> x - y = mq and y - z = mq' for some q, q' ∈ I
=>(x-y)+(y-z) = m(q+q')
=> x - z = m(q + q'), q + q' ∈ I
(x,z) ∈ R
=> R is transitive.
Hence the relation is equivalence relation.

Which of the following sets are null sets ?
  • a)
    {0}
  • b)
    ø
  • c)
    { }
  • d)
    Both (b) & (c)
Correct answer is option 'D'. Can you explain this answer?

Vivek Patel answered
There are some sets that do not contain any element at all. For example, the set of months with 32 days. We call a set with no elements the null or empty set. It is represented by the symbol { } or Ø.

"n/m" means that n is a factor of m, then the relation T is
  • a)
    relexive and symmetric
  • b)
    transitive and symmetric
  • c)
    relexive, transitive and symmetric
  • d)
    relexive, transitive and not symmetric
Correct answer is option 'D'. Can you explain this answer?

Anaya Patel answered
′/′ is reflexive since every natural number is a factor of itself that in n/n for n∈N.
′/′ is transitive if n is a factor of m and m is a factor of P, then n is surely a factor of P.
However, ′/′ is not symmetric.
example, 2 is a factor of 4 but 4 is not a factor of 2. 

The number of elements in the Power set P(S) of the set S = {{1,2}, {2,3}, {2,4}} is given by
  • a)
    2
  • b)
    4
  • c)
    8
  • d)
    none of these
Correct answer is option 'C'. Can you explain this answer?

Rohini S. answered
No of elements in a Power set = 2^N       (where N is the no. of elements in a set)
                                                 = 2^3
                                                  =8

In a class of 50 students, 70% students pass in QA and 60% pass in RC. What is the minimum percentage of students who pass in both the papers?
  • a)
    30%
  • b)
    40%
  • c)
    25%
  • d)
    50%
Correct answer is option 'A'. Can you explain this answer?

Gargi Kulkarni answered
We know that n (A∪B ) = n(A ) + n(B) - n(A∩B) 
Assuming that n(Q) = number of students passing QA and n(R) = number of students passing RC.
Hence, n(Q∪R ) = n(Q) + n(R) - n(Q∩R)
Or, n(Q∩R) = n(Q) + n(R) - n(Q∪R) = 70% + 60% - n(Q∪R)
Now the minimum value of n(Q∩R) will occur for the maximum value of n(Q∪R).
The maximum possible value of n(Q∪R) = 100%
So, the minimum value of n(Q∩R) = 30%
 

The number of elements in the power set of the set {{a, b}, c} is
  • a)
    8
  • b)
    4
  • c)
    3
  • d)
    7
Correct answer is option 'B'. Can you explain this answer?

Aarav Sharma answered
Solution:

The power set of a set is the set of all subsets of that set, including the empty set and the set itself.

The given set is {{a, b}, c}.

To find the power set, we need to consider all possible combinations of elements.

Step 1: Find all possible subsets of the set {a, b}

- The subsets of {a, b} are { }, {a}, {b}, {a, b}

Step 2: Add c to each of the above subsets

- { } U {c} = {c}
- {a} U {c} = {a, c}
- {b} U {c} = {b, c}
- {a, b} U {c} = {a, b, c}

Therefore, the power set of the set {{a, b}, c} is { { }, {c}, {a}, {b}, {a, c}, {b, c}, {a, b}, {a, b, c} }.

Hence, the number of elements in the power set is 4.

Therefore, the correct answer is option B.

Order of the power set of a set of order n is
  • a)
    n
  • b)
    2n
  • c)
    n2
  • d)
    2n
Correct answer is option 'D'. Can you explain this answer?

Rcha answered
It's a standard (proven) equation.You can find explanation in +2 Book.Honestly I don't remember the way.

At    the    birthday    party    of    Sherry,    a    baby    boy,    40    persons    chose    to    kiss   him    and    25    chose    to    shake    hands with    him.    10    persons    chose    to    both   kiss    him    and    shake    hands    with    him.    How    many    persons    turned    out    at the    party?
 
  • a)
    35
  • b)
    75
  • c)
    55
  • d)
    25
Correct answer is option 'C'. Can you explain this answer?

Prisha Shah answered
Explanation: 
From the figure, it is clear that the number of  people at  the party were,
30 + 10 + 15 = 55.
We can of course solve this mathematically as below:
Let  n(A) = No. of persons who kissed Sherry = 40
n(B) = No. of persons who shake hands with Sherry = 25
and n(A > B) = No. of   persons who shook  hands with Sherry and kissed him    both =10

Then using  the formula, n(A > B) = n(A) + n(B) -n(A > B)
n(A > B) = 40 + 25 - 10 = 55
Hence, correct answer is 55.
You can know about Important Formulas related to Set Theory through the document:  

In a class of 50 students, 70% students pass in QA, 90% pass in EU and 60% pass in RC. What is the minimum percentage of students who pass in all the papers?
 
  • a)
    15%
  • b)
    30%
  • c)
    40%
  • d)
    20%
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
Analysis:
To find the minimum percentage of students who pass in all the papers, we need to consider the students who passed in each individual paper and find the common percentage among them.

Given Information:
- Total number of students in the class = 50
- Percentage of students who pass in QA = 70%
- Percentage of students who pass in EU = 90%
- Percentage of students who pass in RC = 60%

Calculating the Number of Students:
- Number of students who pass in QA = (70/100) * 50 = 35
- Number of students who pass in EU = (90/100) * 50 = 45
- Number of students who pass in RC = (60/100) * 50 = 30

Calculating the Minimum Percentage:
To find the minimum percentage, we need to find the common percentage among the students who passed in all three papers.

Step 1:
We can start by assuming that all the students who passed in QA also passed in EU and RC. So, the number of students who passed in all three papers is 35.

Step 2:
Now, let's check if all the students who passed in EU also passed in RC. The number of students who passed in EU is 45, but the number of students who passed in both EU and RC is only 30. This means that there are 15 students who passed in EU but failed in RC.

Step 3:
Since the number of students who passed in all three papers is 35 and there are 15 students who passed in EU but failed in RC, the maximum number of students who passed in all three papers is 35 - 15 = 20.

Step 4:
Finally, we can calculate the minimum percentage of students who passed in all three papers by dividing the number of students who passed in all three papers (20) by the total number of students in the class (50) and multiplying by 100.

Minimum percentage = (20/50) * 100 = 40%

Therefore, the minimum percentage of students who pass in all the papers is 40%.

Answer:
The correct answer is option C) 40%.

Directions for Questions 1 to 3:    
Refer to the data  below and answer  the questions that  follow.
Last year, there were 3 sections in the Catalyst, a mock CAT paper. Out of them 33  students cleared the cut-off in Section 1, 34 students cleared the cut-off in Section 2 and 32 cleared the cut-off in Section 3. 10 students cleared the cut-off in Section 1 and Section 2, 9 cleared the cut-off in Section 2 and Section 3, 8 cleared the cut-off in Section 1 and Section 3. The number of people who cleared each section alone wasequal and was 21 for each section.
How many cleared all the three sections?
  • a)
    3
  • b)
    6
  • c)
    5
  • d)
    7
Correct answer is option 'B'. Can you explain this answer?

Sagar Sharma answered
Given information:
- There were 3 sections in the Catalyst paper.
- 33 students cleared the cut-off in Section 1.
- 34 students cleared the cut-off in Section 2.
- 32 students cleared the cut-off in Section 3.
- 10 students cleared the cut-off in both Section 1 and Section 2.
- 9 students cleared the cut-off in both Section 2 and Section 3.
- 8 students cleared the cut-off in both Section 1 and Section 3.
- The number of people who cleared each section alone was equal and was 21 for each section.

Analysis:
To find the number of students who cleared all three sections, we need to analyze the given information and determine the overlapping sections.

Step 1: Determine the number of students who cleared two sections.
- 10 students cleared both Section 1 and Section 2.
- 9 students cleared both Section 2 and Section 3.
- 8 students cleared both Section 1 and Section 3.

Step 2: Determine the number of students who cleared only one section.
- The number of students who cleared each section alone was 21.

Step 3: Calculate the total number of students who cleared at least one section.
- Number of students who cleared Section 1 = 33
- Number of students who cleared Section 2 = 34
- Number of students who cleared Section 3 = 32

By adding the number of students who cleared each section alone, we can calculate the total number of students who cleared at least one section:

Total = (Number of students who cleared Section 1) + (Number of students who cleared Section 2) + (Number of students who cleared Section 3) - (Number of students who cleared two sections) - (Number of students who cleared three sections)

Total = 33 + 34 + 32 - 10 - 9 - 8

Step 4: Calculate the number of students who cleared all three sections.
- The number of students who cleared all three sections = Total - (Number of students who cleared each section alone)

Number of students who cleared all three sections = Total - (21 + 21 + 21)

Solution:
Substituting the values into the equation, we have:

Number of students who cleared all three sections = (33 + 34 + 32 - 10 - 9 - 8) - (21 + 21 + 21)

Number of students who cleared all three sections = 6

Therefore, the correct answer is option B) 6.

The binary relation S =  Φ (empty set) on set A = {1, 2,3}  is
  • a)
    neither reflexive nor symmetric
  • b)
    symmetric and relexive
  • c)
    transitive and reflexive
  • d)
    transitive and symmetric
Correct answer is option 'D'. Can you explain this answer?

Sameer Rane answered
Option D is correct.
  • Reflexive : A relation is reflexive if every element of set is paired with itself. Here none of the element of A is paired with themselves, so S is not reflexive.
  • Symmetric : This property says that if there is a pair (a, b) in S, then there must be a pair (b, a) in S. Since there is no pair here in S, this is trivially true, so S is symmetric.
  • Transitive : This says that if there are pairs (a, b) and (b, c) in S, then there must be pair (a,c) in S. Again, this condition is trivially true, so S is transitive.
Set A is Irreflexive, Symmetric, Anti Symmetric, Asymmetric, Transitive.
But it is not Reflexive.
Thus, option (D) is correct.

In a class of 50 students, 70% students pass in QA, 90% pass in EU, 85% pass in DI and 60% pass in RC. What is the minimum percentage of students who pass in all the papers?
  • a)
    3%
  • b)
    4%
  • c)
    5%
  • d)
    8%
Correct answer is option 'C'. Can you explain this answer?

Aarav Sharma answered
Solution:

To find out the minimum percentage of students who pass in all the papers, we need to find out the intersection of all the percentages.

QA = 70%
EU = 90%
DI = 85%
RC = 60%

Step 1: Find out the percentage of students who passed in at least one subject.

Percentage of students who passed in at least one subject = QA + EU + DI + RC - (Sum of any two) - (Sum of three) + (All four)

= 70 + 90 + 85 + 60 - (70 + 90) - (70 + 85 + 90) + (70 + 85 + 90 + 60)

= 305%

Step 2: Find out the percentage of students who passed in all four subjects.

Let the percentage of students who passed in all four subjects be x%.

So, we have

x% + Percentage of students who passed in at least one subject = 100%

Or, x% + 305% = 100%

Or, x% = 100% - 305%

Or, x% = -205%

Since x% cannot be negative, we can conclude that the percentage of students who passed in all four subjects is 0% (or more than 0%, but less than the minimum passing percentage in any of the four subjects).

Step 3: Find out the minimum percentage of students who pass in all the papers.

As we know that the intersection of all the percentages cannot be less than 0%, we can conclude that the minimum percentage of students who pass in all the papers is greater than 0%.

The options are 3%, 4%, 5%, and 8%.

Out of these options, the minimum percentage of students who pass in all the papers is 5%, which is the correct answer.

Therefore, the correct answer is option (c) 5%.

Directions: 
In a locality having 1500  households, 1000 watch Zee TV, 300 watch NDTV and 750 watch Star Plus.
Based on this information answer the questions that follow:
The minimum number of  households watching both Zee TV and NDTV is:
 
  • a)
    250
  • b)
    300
  • c)
    450
  • d)
    0
Correct answer is option 'D'. Can you explain this answer?

Shivam Malik answered
In this case,the number of households watching ZeeTV and NDTV can be separate from each other since there is no interference required between the households     watching ZeeTV and the households watching NDTV  as their individual sum(1000 +300) is smaller than the 1500 available householdsin the locality.Hence,the answer in this question is 0.

How many numbers from 1 to 100 are not divisible by either 2 or 4 or 5?
  • a)
    30
  • b)
    40
  • c)
    25
  • d)
    45
Correct answer is option 'B'. Can you explain this answer?

Rajdeep Ghoshal answered
Let us first understand the meaning of the statement given in the question—
It is given that the numbers are from 1 to 100, so while counting we will include both the limits, i.e., 1 and 100. Had this been “How many numbers in between 1 to 100 are..." then we would not have included either 1 or 100.
Now to solve this question, we will first find out the number of numbers from 1 to 100 which are divisible by either 2 or 5 (since all the numbers which are not divisible by 2 will not be divisible by 4 also, so we do not need to find the numbers divisible by 4). And then we will subtract this from the total number of numbers i.e., 100. It can be seen below:
Total number of numbers = numbers which are divisible + numbers which are not divisible
So, n(2U5) = n(2) + n(5) - n (2∩5)
Now, n(2) = 50 
n(5) = 20 
n(2∩5) = 10
n(2U5) = 50 + 20 - 10 = 60
Numbers which are not divisible = total number of numbers-numbers which are divisible
=100 - 60 = 40

Directions: 
In a locality having 1500  households, 1000 watch Zee TV, 300 watch NDTV and 750 watch Star Plus.
Based on this information answer the questions that follow:
The maximum number of  households who watch neither of the three channels is:
  • a)
    600
  • b)
    450
  • c)
    500
  • d)
    400
Correct answer is option 'C'. Can you explain this answer?

Aarav Sharma answered
Given Information:
- Total number of households = 1500
- Number of households watching Zee TV = 1000
- Number of households watching NDTV = 300
- Number of households watching Star Plus = 750

To Find:
The maximum number of households who watch neither of the three channels.

Solution:
To find the maximum number of households who watch neither of the three channels, we need to subtract the number of households watching each channel from the total number of households.

Step 1: Find the number of households watching all three channels (Zee TV, NDTV, and Star Plus).
- Zee TV viewers = 1000
- NDTV viewers = 300
- Star Plus viewers = 750

To find the number of households watching all three channels, we need to find the intersection of the three sets: Zee TV viewers, NDTV viewers, and Star Plus viewers.

Number of households watching all three channels = Zee TV viewers ∩ NDTV viewers ∩ Star Plus viewers

Since there is no information given about the overlap between the viewers of each channel, we can assume that there are no households that watch all three channels. Therefore, the number of households watching all three channels is 0.

Step 2: Find the number of households watching at least one of the three channels.
To find the number of households watching at least one of the three channels, we need to find the union of the three sets: Zee TV viewers, NDTV viewers, and Star Plus viewers.

Number of households watching at least one channel = Zee TV viewers ∪ NDTV viewers ∪ Star Plus viewers

Number of households watching at least one channel = Zee TV viewers + NDTV viewers + Star Plus viewers - Number of households watching all three channels

Number of households watching at least one channel = 1000 + 300 + 750 - 0 = 2050

Step 3: Find the number of households watching neither of the three channels.
To find the number of households watching neither of the three channels, we subtract the number of households watching at least one channel from the total number of households.

Number of households watching neither of the three channels = Total number of households - Number of households watching at least one channel

Number of households watching neither of the three channels = 1500 - 2050 = -550

Conclusion:
The maximum number of households who watch neither of the three channels is 500 (Option C).

A survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see which of the three popular options—air conditioning, radio and power windows—were already installed. The survey found:
15 had air conditioning
2 had air conditioning and power windows but no radios
12 had radio
6 had air conditioning and radio but no power windows
11 had power windows
4 had radio and power windows
3 had all three options
What is the number of cars that had none of the options?
  • a)
    4
  • b)
    3
  • c)
    1
  • d)
    2
Correct answer is option 'D'. Can you explain this answer?

Aniket Menon answered
According to the question, it is given that there was a survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see which of the three popular options- air-conditioning, radio, and power windows- were already installed.
15 cars had to air-condition. This doesn’t mean that it only had to air-condition. It can also have a radio, or power-windows, or both.
2 cars had air-conditioning and power windows but no radios.
12 cars had power windows. This doesn’t mean that it only had power windows. It can also have a radio, or air-conditioning, or both.
6 cars had air-conditioning and radio but no power windows.
11 cars had a radio. This doesn’t mean that it only had a radio. It can also have power-windows, or air-conditioning, or both.
4 cars had radio and power windows. It may or may not have to air-condition.
3 cars had all three options.

Now, on drawing Venn diagram, we get
When we add all the values, we get a total of 23 cars. 
So, 2 cars don’t have the air conditioning or the radio, or the power windows.

If  f : X -> Y and a, b  ⊆ X, then f (a  ∩ b) is equal to
  • a)
    f(a) - f(b)
  • b)
    f(a) ∩ f(b)
  • c)
    a proper subset of f(a) ∩ f(b)
  • d)
    f(b) - f(a)
Correct answer is option 'C'. Can you explain this answer?

Arya Roy answered
The only requirement to answer the above question is to know the definition of function- a relation becomes a function if every element in domain is mapped to some element in co-domain and no element is mapped to more than one element.

Now, we have a,b⊆ X. Their intersection can be even empty set. So, lets try out options:

Options a and d don't even need a check.

Lets take a case where a∩b = φ. Now, f(a∩b)=φ, but f(a) ∩ f(b) can be non empty. So, option B can be false.

Option C is always true provided "proper subset" is replaced by "subset". This is because no element in domain of a function can be mapped to more than one element. And the subset needn't be "proper" as for a one-one mapping, we get
f(a∩b) = f(a) ∩ f(b) ,Hence option C

Directions: 
In a locality having 1500  households, 1000 watch Zee TV, 300 watch NDTV and 750 watch Star Plus.
Based on this information answer the questions that follow:
The minimum number of households watching Zee TV and Star Plus is:
  • a)
    300
  • b)
    250
  • c)
    275
  • d)
    350
Correct answer is option 'B'. Can you explain this answer?

Madhurima Dey answered
If we try to consider each of the households watching Zee TV and   StarPlus as independent of  each   other, we would get a total of 1000 + 750 = 1750 households.However, wehave a totalof only1500 householdsi the locality and hence, there has to be a minimum interference of atleast 250 households whowould  be watching both Zee TV and Star Plus. Hence, the answer to this question is 250.

Last year, there were 3 sections in the Catalyst, a mock CAT paper. Out of them 33  students cleared the cut-off in Section 1, 34 students cleared the cut-off in Section 2 and 32 cleared the cut-off in Section 3. 10 students cleared the cut-off in Section 1 and Section 2, 9 cleared the cut-off in Section 2 and Section 3, 8 cleared the cut-off in Section 1 and Section 3. The numberof people who cleared each section alone wasequal and was 21 for each section.
The ratio of  the number of students clearing then cut-off in one or  more of  the sections to     the number of  students clearing the cutoff  in Section 1 alone is?
  • a)
    78/21
  • b)
    3
  • c)
    73/21
  • d)
    none of these
Correct answer is option 'A'. Can you explain this answer?

Madhurima Dey answered
(21+ 21 +21 +6+4+3+2)/21 = 78/21
Option (A) is correct
 

The set of all Equivalence classes of a set A of cardinality C
  • a)
    has the same cardinality as A
  • b)
    forms a partition of A
  • c)
    is of cardinality 2C
  • d)
    is of cardinality C2
Correct answer is option 'B'. Can you explain this answer?

Meera Rana answered
The equivalence classes of any equivalence relation on A partition A. There's no need to talk about cardinalities to know this; that's just the fact that equivalence relations are equivalent to partitions.

If A is the natural numbers, and we take just one equivalence class (all of A), then a), b), d) claim that there are infinitely many equivalence classes. But there's just one.

In a group of 120  athletes, the number of athletes who can  play Tennis, Badminton, Squash and Table Tennis is 70, 50, 60 and 30 respectively. What is the maximum number   of athletes who can play none of the games
  • a)
    40
  • b)
    50
  • c)
    60
  • d)
    70
Correct answer is option 'B'. Can you explain this answer?

Rajdeep Ghoshal answered
In order to think of the maximum   number of  athletes who can play   none of  the games, we can think   of  the fact that since there  are 70 athletes who play tennis, fundamentally there are a maximum of 50 athletes who would be possibly in the can play none of the games’.
No other constraint in the problem necessitates a reduction of this number and hence the answer to this question is 50.

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Mathematics for JAMB

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