De Morgan's Law states that (A ∪ B)' is equal to:
Answer: B
De Morgan's Laws are fundamental principles in set theory.
The first law states that the complement of the union of two sets is the intersection of their complements: (A ∪ B)' = A' ∩ B'.
The second law states that the complement of the intersection of two sets is the union of their complements: (A ∩ B)' = A' ∪ B'.
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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.
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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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