If A and B are disjoint sets, then n(A ∩ B) is:
Answer: A
Disjoint sets share no elements at all, so their intersection is empty.
\[ n(A \cap B) = 0 \]
This is why \( n(A \cup B) = n(A) + n(B) \) for disjoint sets, with no correction term.
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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.
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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.
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In a class of 120 students, 90 students passed in History, 65 students passed in Sociology and 30 students failed in both the subjects. How many students passed in both the subjects?
Answer: C
Total students = 120.
Number of students who failed in both subjects = 30.
Number of students who passed in at least one subject = Total students - Failed in both = 120 - 30 = 90.
So, n(H ∪ S) = 90.
Given n(H) = 90 and n(S) = 65.
Using the formula n(H ∪ S) = n(H) + n(S) - n(H ∩ S):
90 = 90 + 65 - n(H ∩ S)
n(H ∩ S) = 155 - 90 = 65.
So, 65 students passed in both subjects.
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For any two sets A and B, A ∩ (A ∪ B) is equal to:
Answer: A
This is an absorption law in set theory.
The set A ∪ B contains all elements of A plus all elements of B.
When we take the intersection of this larger set with A, we are left with only the elements that are in A.
Therefore, A ∩ (A ∪ B) = A.
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In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?
Answer: B
Let H be the set of students who know Hindi and E be the set of students who know English.
n(H) = 100, n(E) = 50, n(H ∩ E) = 25.
Since each student knows at least one language, the total number of students is the union of the two sets, n(H ∪ E).
n(H ∪ E) = n(H) + n(E) - n(H ∩ E) = 100 + 50 - 25 = 125.
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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.
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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.
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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.
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If A = {1, 2, {3, 4}}, which of the following is a true statement?
Answer: B
The elements of set A are 1, 2, and the set {3, 4}.
Option A is false because 3 is not an element of A; it's an element of the element {3, 4}.
Option B is true because 1 and 2 are elements of A, so the set containing them, {1, 2}, is a subset of A.
Option C is false because {3, 4} is an element of A, not a subset. For it to be a subset, 3 and 4 would have to be elements of A themselves.
Option D is false for the same reason as A.
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