If n(A)=25 and n(B)=40 and n(A∪B)=50, find n(A Δ B).
Answer: C
The symmetric difference A Δ B is the set of elements in exactly one of the sets.
First, find the intersection: n(A∩B) = n(A)+n(B)-n(A∪B) = 25+40-50=15.
Now, n(A Δ B) = n(A∪B) - n(A∩B) = 50 - 15 = 35.
Alternatively: n(A only) = 25-15=10. n(B only) = 40-15=25. Total = 10+25=35.
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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}
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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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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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If n(U) = 700, n(A) = 200, n(B) = 300 and n(A ∩ B) = 100, then n(A' ∩ B') is:
Answer: C
Using De Morgan's Law, we know that A' ∩ B' = (A ∪ B)'.
Therefore, n(A' ∩ B') = n((A ∪ B)') = n(U) - n(A ∪ B).
First, we find n(A ∪ B):
n(A ∪ B) = n(A) + n(B) - n(A ∩ B) = 200 + 300 - 100 = 400.
Now, n(A' ∩ B') = n(U) - 400 = 700 - 400 = 300.
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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.
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In a school, there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach both physics and mathematics. How many teach physics?
Answer: B
Let M be the set for Math teachers and P for Physics teachers.
n(M ∪ P) = 20, n(M) = 12, n(M ∩ P) = 4.
We use the formula n(M ∪ P) = n(M) + n(P) - n(M ∩ P).
20 = 12 + n(P) - 4
20 = 8 + n(P)
n(P) = 12.
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In a school, 300 students play cricket and 250 play hockey. If 110 students play both games, how many students play either cricket or hockey?
Answer: B
Let C be the set for cricket and H for hockey.
n(C) = 300, n(H) = 250, n(C ∩ H) = 110.
The number of students who play either game is the union, n(C ∪ H).
n(C ∪ H) = n(C) + n(H) - n(C ∩ H)
n(C ∪ H) = 300 + 250 - 110 = 550 - 110 = 440.
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The Idempotent Law for set union states that:
Answer: A
The Idempotent Law states that when an operation is applied to an element with itself, the result is the same element.
For set theory, this means:
1. A ∪ A = A (Idempotent Law of Union)
2. A ∩ A = A (Idempotent Law of Intersection)
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Let U = {1, 2, 3, 4, 5, 6, 7, 8} be the universal set and A = {1, 3, 5, 7}. Find A'.
Answer: C
The complement of a set A, denoted A', consists of all elements in the universal set U that are not in A.
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 3, 5, 7}
A' = {2, 4, 6, 8}
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