Test: Sets- 2 - GMAT MCQ

Concept use:

A - B = The element which are present in A but not in B

Calculations:

If set A = {1, 3, 5, 7}, set B = {1, 4, 7}, 

The Value of A - B = {3, 5} (The element which are present in A but not in B)

Given:

Cartesian product of two sets A and B: A × B = {(3, 2), (3, 4), (5, 2), (5, 4)}

Concept used: 

In a Cartesian product A × B, the first element of each ordered pair belongs to set A, and the second element belongs to set B.

Calculation:

From the given Cartesian product A × B, we can extract the elements of set A:

A = {3, 5}

∴ The set A is {3, 5}.

Given:
The expression is A ∪ (B ∩ C)
We use the concept of set theory.
According to the distributive laws of sets, A ∪ (B ∩ C) is equivalent to (A ∪ B) ∩ (A ∪ C).
∴ The correct option is (D) (A ∪ B) ∩ (A ∪ C).

Given that

A = {2, 3, 4}

B = {3, 5}

Union of A and B 

A ∪ B = {2, 3, 4, 5}

Therefore, A ∪ B = {2, 3, 4, 5}.

CONCEPT:

Let A and B be two sets then A is said to be proper subset of B, if A is a subset of B and A is not equal to B. It is denoted as A ⊂ B.

CALCULATION:

Given: A = {1, 3, 4} and B = {x : x ∈ R and x² - 7x + 12 = 0}

First let's find the roaster form of set B

In order to do so we need to find the roots of the equation x² - 7x + 12 = 0

⇒ x² - 3x - 4x + 12 = 0

⇒ x(x - 3) - 4(x - 3) = 0

⇒ (x - 4) × (x - 3) = 0

⇒ x = 3, 4

⇒ B = {3, 4}

As we can clearly see that, all the elements of B are there in set A but A ≠ B i.e B ⊂ A

Hence, the correct option is 3.

Concept:

Combination: Selecting r objects from given n objects.

• The number of selections of r objects from the given n objects is denoted by  ⁿC_r
• 

Calculation:

Number of elements in A = 10

Number of subsets of A containing exactly two elements = Number of ways we can select 2 elements from 10 elements

⇒ Number of ways we can select 2 elements from 10 elements = ¹⁰C₂ = 45

∴ Number of subsets of A containing exactly two elements = 45

CALCULATION:

Given: A = {1, 2, 3, 4, 5, 7, 8, 9} and B = {2, 4, 6, 7, 9}

As we know that, A ∩ B = {x : x ∈ A and x ∈ B}

⇒ A ∩ B = {2, 4, 7, 9}

As we can see that,

The number of elements present in A ∩ B = 4

i.e n(A ∩ B) = 4

As we know that;

If A is a non-empty set such that n(A) = m then

The numbers of proper subsets of A are given by 2^m - 1.

So, The number of proper subsets of A ∩  B = 2⁴ - 1 = 15

Hence, the correct option is A

CALCULATION:

Given: A = {1, 2, 5, 7} and B = {2, 4, 6}

As we know that,  A ∪ B = {x : x ∈ A or x ∈ B}.

⇒ A ∪ B = {1, 2, 4, 5, 6, 7}

As we can see that, the number of elements present in A U B = 6 i.e n(A U B) = 6

As we know that, if A is a non-empty set such that n(A) = m then numbers of proper subsets of A is given by 2m - 1.

So, the number of proper subsets of A Δ B = 2⁶ - 1 = 63

Hence, the correct option is 3.

CALCULATION:

Given: A = {1, 3, 5}

Here, we have to find the cardinality of the power set of A i.e n (P(A))

As we know that if A is a finite set with m elements. Then the number of elements (cardinality) of the power set of A is given by: n (P(A)) = 2m.

Here, we can see that, the given A has 3 elements i.e n(A) = 3

So, the cardinality of the given set is n(P(A)) = 23 = 8

Hence, the correct option is 4.

Given, A = {x, y, z}. 

The power set (or powerset) of a Set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set.

Powerset of A = {ϕ,{x}, {y}, {z}, {x, y}, {y, z}, {x, z},{x, y, z}}.

Hence, the number of subsets in the powerset of A is 8.

We divide the set into n classes {1, 2}, {3, 4},......{2n - 1, 2n}.

By the pigeonhole principle, given n +1 elements at least two of them will be in the same case {2k - 1, 2k} (1 ≤ k ≤ n). But 2k - 1 and 2k are relatively prime because their difference is 1. 

Calculations:

Since, A ⊂ B and B ⊂ C, therefore A ∪ B ∪ C = C.

⇒ n(A ∪ B ∪ C) = n(C).

∴ The cardinality (number of elements) of A ∪ B ∪ C = cardinality (number of elements) of C.

Given:

Total consumers = 1000

Consumers who like product A = 720 

Consumers who like product B = 450

Formula Used: 

n(A U B) = n(A) + n(B) - n(A ∩ B) 

Calculation:

Least number that like both the products are n(AꓵB)

⇒ 1000 = 720 + 450 - n(A ∩ B)   

⇒ 1000 = 1170 - n(A ∩ B)

⇒ n(A ∩ B) = 170

∴ 170 consumers like both the products A and B 

Concept:

Equivalent set: Two sets A and B are said to be equivalent if they have the same number of elements (cardinality), regardless of what the elements are. , i.e n(A) = n(B).

Equal set: Two sets A and B are said to be equal if they have exactly the same elements. The repetition of elements does not matter in sets because a set is defined as a collection of distinct objects, so repeated elements are generally ignored.

If A = B, then n(A) = n(B) and for any x ∈ A, x ∈ B too.

Calculation:

From the definition of equivalent and equal set we can see that;

Statement 1 is true because the number of elements in A = Number of elements in B = 3

Statement 2 is true because {1, 5, 9} ∈ A and {1, 5, 9} ∈ B

Let the number of rural residents who own a car = p
STATEMENT 1: The number of rural residents in the group is 60% of the number of urban residents in the group.
Let the number of urban residents = x
Then, number of rural residents = 0.6x
We know that, total number of people = number of people who own a car + number of people who don't own a car = number of urban residents + rural residents
total = (15 + p) + 18 (given in the question) = x + 0.6x
We have two variables x and p but only 1 equation.
Insufficient
STATEMENT 2: The difference between the number of urban residents who own a car and the number of rural residents who own a car is one-fifth of the number of rural and urban residents in the group.
Two equations can be formed here:
1) 15 - p = 1/5(rural + urban) or
2) p-15 = 1/5(rural + urban)
Both will give different answers of p
Insufficient
COMBINING BOTH STATEMENTS:
1) 15 - p = 1/5 (rural + urban)
75 - 5p = rural + urban
75 - 5p = 0.6x + x
75 - 5p = 15 + p + 18 (from statement 1)
42 = 6p
p = 7
2) p - 15 = 1/5 (rural + urban)
5p - 75 = 15 + p + 18
4p = 108
p = 27
We have two values of p.
Let's put it in equation 1 and get the value of x (the number of urban residents)
For p = 7,
15 + p + 18 = 1.6 x
33 + 7 = 1.6x
x = 25
For p = 27,
15 + p + 18 = 1.6 x
33 + 27 = 1.6x
1.6 x = 60
x = 75/2 = 37.5 INVALID
The value of x represents number of people and cannot be a fraction. Therefore, there is only one unique value of x and that is x = 25 when p = 7
Thus, (C) is sufficient.

Steps 1 & 2: Understand Question and Draw Inferences

Given:

To find:

• Let number of girls who do not like history = X
• Total number of girls = Y
• We need to find the value of X/Y
•  to answer the question
• Completing the above table in terms of X and Y:

Step 3: Analyze Statement 1 independently

One-third of the boys in the class like to study History

Sufficient

Step 4: Analyze Statement 2 independently

The number of girls who like History is 50 percent of the number of girls who do not like History

Sufficient

Step 5: Analyze Both Statements Together (if needed)

Since we’ve already arrived at a unique answer in Step 3 and Step 4, this step is not required

Answer: Option D

Given:

• Percentage of employees who got a promotion = 40%
• Percentage of employees who got a salary hike = 50%
• Percentage of employees who neither got a promotion nor a salary hike = 30%
• Number of employees who got both the promotion and the salary hike = 40

To Find: Number of employees in the company?
Approach:

To find the total number of employees in the company, we will try to express the number of employees who got both the promotion and the salary hike(i.e. 40), as a percentage of the total employees of the company.

Total employees in the company = Employees who got only salary hike + Employees who got only promotion + Employees who got salary hike and promotion + Employees who got neither salary hike nor promotion

1. Adding “Employees who got salary hike and promotion on both sides” we have
2. Total employees in the company + Employees who got salary hike and promotion = (Employees who got only salry hike + Employees who got salary hike and promotion)+ (Employees who got only promotion + Employees who got salary hike and promotion) + Employees who got neither salary hike nor promotion
3. Total employees in the company + Employees who got salary hike and promotion = Employees who got a salary hike+  Employees who got a promotion  + Employees who got neither salary hike nor promotion
4. From the above equation, we can find the percentage of employees who got salary hike and promotion

Working Out:

1. Total employees in the company + Employees who got salary hike and promotion = Employees who got a salary hike+  Employees who got a promotion  + Employees who got neither salary hike nor promotion
   a.  100% + Employees who got salary hike and promotion = 50% + 40% + 30%
   b.  Employees who got salary hike and promotion = 20%
2. Now, we know that employees who got salary hike and promotion = 40
   1. So, if 20% is 40, 100% = 200

Hence, the total number of employees in the company = 200
Answer: D

Given:

• Number of employees = 100
• Let us assume:
  • Number of employees who only have an MBA degree = c.
  • Number of employees who have both an MBA and an Engineering degree = b.
  • Number of employees who do neither have an MBA nor an Engineering degree = d.
• Employees having neither Engineering nor an MBA were the least in number.
  • So d is the least in the number out of 20,b, c & d.
• Number of employees who had an Engineering degree and not an MBA degree = 20

To Find: Among the options, which of them cannot be the number of employees who had an MBA degree?
Approach:

1. Let’s first draw the venn diagram to understand the information given

2. We need to find the possible range of values of employees who had an MBA degree = b + c

• So, we need to find the maximum and minimum value of b + c

3. As we know that total number of employees is 100 and d is minimum number of employees possible, we will use this information to find out the range of values of b + c
Working Out:

1. As the total number of employees = 100
   • 20 + b + c + d = 100
   • b + c + d = 80
2. Maximum value of b + c
   • Maximum value of b + c is possible, when d is minimum
   • As the minimum value of d = 0, maximum value of b + c = 80…….(1)
3. Minimum value of b + c
   • b + c will be minimum , when d is maximum
   • Since we know the number of employees who had only an Engineering degree = 20, d < 20
   • So, maximum value of d = 19 (remember that out of 20, b, c and d, we are given that d holds the minimum value. So, d must be less than 20)
   • Hence, minimum value of b + c = 80 – 19 = 61…….(2)
4. Using (1) and (2), we can write 61 ≤ b + c ≤ 80
5. The only value in the options that does not lie in the range is 55, option A

Answer: A

Given

• Number of employees = 210
• Number of employees who did not speak English = 60
  • So, number of employees who spoke English = 210 – 60 = 150
• Number of employees who did not speak German = 150
  • So, number of employees who spoke German = 210 – 150 = 60
• Number of employees who did not speak French = 170
  • So, number of employees who spoke French = 210 – 170 = 40
• Number of employees who did not speak German or English = 20
  • So, these employees must speak only French (because it’s given that each employee speaks at least one language)
• Number of employees who did not speak French and English = 20
  • So, these employees must speak only German (because it’s given that each employee speaks at least one language)

To Find: Maximum number of employees who spoke only English

Approach:

1. To find the maximum number of employees who spoke only English, we will first analyse the relation between total number of employees and number of employees who speak only English.
   • Total number of employees = number of employees (who only speak English + who only speak French + who only speak German + who speak both German & French + who speak both German & English + who speak both French & English + who speak all the three languages + who speak none of the languages)
   • Number of employees who only speak English = (Total number of employees) – (number of employees (who only speak French + who only speak German + who speak both German & French + who speak both German & English + who speak both French & English + who speak all the three languages + who speak none of the languages)
   • As we need to maximize the number of employees who spoke only English, we need to minimize the number of employees who spoke English as well as some other language.
   • So, of the employees who speak more one language we would maximize the number of employees who spoke only French and German
2. Now, we know the number of employees who spoke French and German and also the number of employees who spoke only  French and German.
3. We will use this information to maximize the number of employees who spoke only French and German.

Working Out

1. As there are 40 employees who spoke French and 20 employees who only spoke French,              
   • Number of employees who spoke French and some other language = 40 – 20 = 20
2. Similarly, as there are 60 employees who spoke German and 20 employees who spoke only German,
   • Number of employees who spoke German and some other language = 60 – 20 = 40
3. Total number of employees who spoke more than 1 language
   • Number of employees who spoke English + Number of employees who spoke French + Number of employees who spoke German – Total number of employees = 150 + 60 + 40 – 210 = 40
4. As there are 40 employees who spoke more than 1 language, we will try to distribute maximum number of these employees into the number of employees who spoke French and German only.
   • As there are 20 employees who spoke French and some other language and there are 40 employees who spoke German and some other language, the maximum number of employees who spoke French and German = 20
5. So, that still leaves us with 40 – 20 = 20 employees who spoke German and some other language.
   • The only possibility is for these employees to speak German and English
6. So, there are 20 employees who spoke English and some other language
7. As there are 150 employees who spoke English, there can be a maximum of 150 – 20 = 130 employees who spoke only English.

Answer: C

Given:

• The Women in the group are divided into groups based on 2 parameters:
  • Employment (Employed, Unemployed)
  • Education (Have a Graduate Degree, Don’t have Graduate Degree)
• So, the given information can be represented in a table as follows:

• Now, % of women Unemployed = 100% - 70% = 30%
• And, % of Women with Graduate Degree = 100% - 25% = 75%

To find: Which of the 3 statements cannot be true?

• That is, which of the 3 statements are definitely false

Approach:

1. We’ll evaluate the 3 statements one by one to evaluate which of them are definitely false.

Working Out:

• Evaluating Statement I
• No woman in the group who had a graduate degree was unemployed
  • Let the Women with Graduate Degree who are Unemployed be X, and let the Women with Graduate Degree who are Employed be Y.
  • As per this Statement, X = 0%

• Now, the maximum possible value of Y is 70% (happens when ALL employed women have graduate degrees)
• So, the minimum possible value of X is 5% (since the sum of Y and X is 75%)
• Therefore, Statement I is definitely false.

• Evaluating Statement II
• Less than half of the women in the group were employed and had a graduate degree
  • Again, Let the women who were employed and had Graduate Degree be Y
  • As per this Statement, Y < 50%

Remember that the question is asking if a Statement is definitely false (that is, false for all possible values of Y)

• In our analysis of Statement I above, we saw that the maximum possible value of Y is 70%
  • Therefore, for the maximum value of Y, Statement II is false
  • But we cannot say at this point if Statement II is definitely false (that is, false for all possible values of Y)
• So, let’s now evaluate is also false for the minimum value of Y.
  • The maximum number of Employed Women without Graduate Degree is 25% (happens when ALL women without Graduate Degree are Employed)

​

• So, the minimum value of Y is 70% - 25% = 45%
• We observe that for the minimum value of Y, Statement II is true.
• Thus, Statement II is true for some values of Y and false for other values of Y.
   
• So, Statement II is not definitely false.
• Evaluating Statement III
• The number of unemployed women with a graduate degree was 50 percent greater than the number of employed women without a graduate degree
  • Let the Women with Graduate Degree who are Unemployed be X, and let the Women without Graduate Degree who are Employed be Z.
  • As per Statement III, X = 1.5Z

• Note that this Statement will be definitely false if no valid pair of (X,Z) satisfies the equation X = 1.5Z
• We will first find the values of X and Z that will satisfy this equation as well as the table above. Then, we will check if these values are acceptable.

• So, the equation that we can write to relate X and Z is:
  • X = 75% - (70% - Z)
    • = 5% + Z
  • Let’s now substitute X = 1.5Z in this equation:
    • 1.5Z = 5% + Z
    • So, 0.5Z = 5%
    • Therefore, Z = 10%
    • Corresponding value of X = 1.5*10% = 15%
• So, X = 1.5Z is satisfied by X = 15% and Z = 10%
• We now need to check if these values of X and Z are acceptable
  • The minimum possible value of Z is 0%.
  • In this case, the table will look as under:

The maximum possible value of Z is 25%. In this case, the table will look as under:

• So, the values of Z range from 0% to 25% and correspondingly, the values of X range from 5% to 30%
• So, the values X = 15% and Z = 10% do indeed fall in the acceptable range of values of X and Z.
• So, Statement III is true for one particular pair of X and Z
• Therefore, Statement III is not definitely false

Getting to the answer

• We’ve seen above that out of Statements I, II and III, only Statement I is definitely false.

Looking at the answer choices, we see that the correct answer is Option A

Steps 1 & 2: Understand Question and Draw Inferences

Given:

• Let us assume:
  • a be the number of people who ate only chocolate ice-cream
  • c be the number of people who ate only strawberry ice-cream
  • b be the number of people who ate both chocolare & strawberry ice-cream
  • d be the number of people who did not eat any of ice-creams
• This is represented in the venn diagram below:

• Some people ate both types of ice-cream
  • b > 0 
• Atleast one person did not eat any type of ice-cream
  • d > 0
• Atleat one person ate only one type of ice-cream
  • Either of a or c or both > 0

To Find: If a + b > c + b ?

• If a > c ?

Step 3: Analyze Statement 1 independently

The ratio of the numbers of people who ate chocolate ice-cream and people who ate strawberry ice-cream was greater than the ratio of the total number of people who went to ice-cream parlour and the number of people who did not eat any type of ice-cream.

• Minimum value of a+b+c+d = a+b+d
  • ​Either of a or c >0. So let us take a case, when a>0 and c=0

​​​

• Using (1), we can write:  
• a + b > b + c
• a > c

• Sufficient to answer

Step 4: Analyze Statement 2 independently

The number of people who ate only one type of ice-cream is greater than the number of people who ate strawberry ice-cream.

• a + c > b + c 
• a > b 
• Does not tell us if a > c.

• Insufficient to answer

Step 5: Analyze Both Statements Together (if needed)

As we have a unqiue answer from step-3, this step is not required.

Answer: A

Given

• Number of students in the class = 200
• Number of female students = 80
  • So, number of male students = 200 – 80 = 120
• Number of female non-American students = 10% of 200 = 20
• Number of American students = 3/5 * 200 = 120

To Find: Number of students who are male Americans?
Approach:

1. To find the number of male americans, we will use the relation:

• Number of American students = Number of female American students + Number of male American students             
• We know the number of American students, but we do not know the number of female American students.
• So, we need to find the number of female American students

2. Now, Number of female students = Number of female American students + Number of female non-American students

• As we know the number of female students and the number of female non-American students, we can calculate the the number of female American students

Working Out:

1. Number of female students = 80
   • Number of female non-American students = 20
   • Number of female American students = 80 – 20 = 60
2. Number of American students = 120
   • Number of female American students = 60 (calculated above)
   • So, Number of male American students = 120 – 60 = 60

So, there are 60 male American students.

Answer: C

Given:

• Let the number of people who own only a car be x
  • So, number of people who own only a bike = 3x
  • Also, since x denotes the number of people, x must be a non-negative integer
• Number of people who own a car and a bike = ½ *(Number of people who own either a bike or a only car) = ½ * (x+3x) = 2x
• Number of people who neither own a car nor a bike = Number of people who own a bike

To Find: Among the options, which can be the total people in the locality?

Approach:

1. We will try to express the number of people in terms of x through the venn diagram and then evaluate the possible options.

Working Out:

1. Number of people who own a bike = 3x + 2 x = 5x
   • So, the number of people who own neither a bike nor a car = 5x
2. Hence, total people in the locality = 3x + 2x + x + 5x = 11x
   • So, the total people in the locality should be a multiple of 11
   • The only multiple of 11 in the options is 66, option C

Answer: C

Given:

• Total number of students = 66
• Students who take:
  • Science = 30
  • Humanities = 40
  • Dance = 40
  • None of these 3 subjects = 0
• Maximum possible number of students who study all 3 subjects = x

To find: X – N = ?

Approach:

1. To answer the question, we need to find the value of x
2. We’ll visually represent the given information to analyze the maximum possible overlap for 3 subjects.

Working Out:

The shaded part represents the students who study all three subjects: Science, Humanities, and Dance.

In the problem, the shaded part is labeled as x, which represents the maximum number of students who study all three subjects.

• Analyzing the Maximum possible overlap for the 3 subject

• To maximize x, i.e. to minimize the other overlap regions, among the sets, let’s consider the number of students who study only two subjects to be 0
  • So, number of students who study only Science = 30 – x
  • Number of students who study only Humanities = 40 -x
  • Number of students who study only Dance = 40 – x
• So, number of students who study only 1 subject + Number of students who study all the 3 subjects = 66
• (30-x) + (40-x) + (40- x) + x = 66
• 2x = 44
• x = 22

• Getting to the answer
  • So, the maximum number of students who can study all the 3 subjects is 22

Looking at the answer choices, we see that the correct answer is Option D

Calculation:

Complement of A = (A') = All elements not in A

Complement of A'

((A')') = All elements not in A', which are the elements in A.

Therefore, (A')' = A.