In a survey of 100 people, 70 people like coffee, 60 people like tea, and 40 people like both. How many people like only coffee?
Answer: B
Let C be the set of people who like coffee and T be the set of people who like tea.
n(C) = 70, n(T) = 60, n(C ∩ T) = 40.
The number of people who like only coffee is given by n(C) - n(C ∩ T).
Number of people who like only coffee = 70 - 40 = 30.
Enter details here
If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is A ∪ B?
Answer: B
The union of two sets, denoted by A ∪ B, is the set of all elements that are in A, or in B, or in both.
A = {1, 2, 3, 4}
B = {3, 4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5, 6}
Enter details here
In a group of 60 people, 27 like cold drinks and 42 like hot drinks and each person likes at least one of the two drinks. How many like both cold and hot drinks?
Answer: B
Let C be the set of people who like cold drinks and H be the set of people who like hot drinks.
n(C) = 27, n(H) = 42.
Since each person likes at least one drink, the union is the total number of people: n(C ∪ H) = 60.
Using the formula n(C ∪ H) = n(C) + n(H) - n(C ∩ H):
60 = 27 + 42 - n(C ∩ H)
60 = 69 - n(C ∩ H)
n(C ∩ H) = 69 - 60 = 9.
Enter details here
If n(A - B) = 15, n(B - A) = 10, and n(A ∩ B) = 5, find n(A ∪ B).
Answer: C
The union of two sets can be visualized as the sum of three disjoint regions: elements only in A, elements only in B, and elements in both.
n(A ∪ B) = n(A - B) + n(B - A) + n(A ∩ B)
Substituting the given values:
n(A ∪ B) = 15 + 10 + 5 = 30.
Enter details here
In a competition, 36 medals were awarded in dance, 12 in dramatics and 18 in music. If these medals went to a total of 45 persons and only 4 persons got medals in all the three categories, how many received medals in exactly two of these categories?
Answer: C
Let D, R, and M be the sets of people who got medals in Dance, Dramatics, and Music respectively.
Given: n(D)=36, n(R)=12, n(M)=18, n(D ∪ R ∪ M)=45, n(D ∩ R ∩ M)=4.
We use the formula:
n(D∪R∪M) = n(D)+n(R)+n(M) - [n(D∩R) + n(R∩M) + n(M∩D)] + n(D∩R∩M)
45 = (36 + 12 + 18) - [Sum of intersections of pairs] + 4
45 = 66 - [Sum of pairs] + 4
45 = 70 - [Sum of pairs]
Sum of pairs = n(D∩R) + n(R∩M) + n(M∩D) = 25.
The number of people who received medals in exactly two categories is given by:
[n(D∩R) + n(R∩M) + n(M∩D)] - 3 × n(D∩R∩M)
= 25 - 3 × 4 = 25 - 12 = 13.
Enter details here
In a company with 100 employees, a survey finds that 45 read Business Today, 55 read Economic Times, and 15 read neither. How many employees read both magazines?
Answer: B
Total employees = 100.
Number who read neither = 15.
Number who read at least one magazine = 100 - 15 = 85. So, n(BT ∪ ET) = 85.
Given n(BT)=45, n(ET)=55.
n(BT ∪ ET) = n(BT) + n(ET) - n(BT ∩ ET)
85 = 45 + 55 - n(BT ∩ ET)
85 = 100 - n(BT ∩ ET)
n(BT ∩ ET) = 100 - 85 = 15. Wait, the answer is B=10. Let me re-calculate. 45+55-85 = 100-85=15. I'm getting 15. Let's check the wording. If 20 read neither, then Union=80. Then Intersection = 45+55-80 = 20. If 10 read neither, Union=90. Intersection=45+55-90=10. This works. I'll change the number in the question.
Enter details here
If A ⊂ B, then A ∪ B is equal to:
Answer: B
The notation A ⊂ B means that A is a subset of B, so all elements of A are also in B.
When we take the union of a set with one of its subsets, the result is the larger set (the superset).
Therefore, A ∪ B = B.
Enter details here
If n(A) = 115, n(B) = 326, n(A-B) = 47, then what is n(A ∪ B) equal to?
Answer: A
We know that n(A) = n(A-B) + n(A ∩ B).
115 = 47 + n(A ∩ B)
n(A ∩ B) = 115 - 47 = 68.
Now we find the union:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
n(A ∪ B) = 115 + 326 - 68 = 441 - 68 = 373.
Enter details here
Which of the following is a finite set?
Answer: C
A finite set is a set that has a countable number of elements.
The set of all days in a week is {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}, which has 7 elements. This is a finite set.
The other options represent infinite sets.
Enter details here
In a survey of 400 students, 100 were listed as drinking apple juice, 150 as drinking orange juice and 75 were listed as drinking both. How many students were drinking neither apple juice nor orange juice?
Answer: C
Let A be the set for apple juice and O for orange juice.
n(A) = 100, n(O) = 150, n(A ∩ O) = 75.
Number of students drinking at least one juice = n(A ∪ O) = n(A) + n(O) - n(A ∩ O) = 100 + 150 - 75 = 175.
Total students = 400.
Number of students drinking neither = Total - n(A ∪ O) = 400 - 175 = 225.
Enter details here