In a group of 80 people, 35 speak only Hindi and 20 speak only English. If everyone speaks at least one language, how many speak both?
Answer: C
The total number of people is the sum of those who speak only Hindi, only English, and both.
Total = (Only Hindi) + (Only English) + (Both)
80 = 35 + 20 + (Both)
80 = 55 + (Both)
Both = 80 - 55 = 25.
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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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Given three sets A, B, C. Which expression represents the elements that are in C, but not in A or B?
Answer: A
We want elements that are in C and are not in A and not in B.
The set of elements in A or B is A ∪ B.
To find the elements in C that are not in this union, we perform a set difference.
The correct expression is C - (A ∪ B).
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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
Correct answer: 15
If 15 of the 100 employees read neither magazine, then the number reading at least one is
\[n(BT \cup ET) = 100 - 15 = 85\]
Apply the inclusion–exclusion principle:
\[n(BT \cup ET) = n(BT) + n(ET) - n(BT \cap ET)\]
\[85 = 45 + 55 - n(BT \cap ET) \quad\Longrightarrow\quad n(BT \cap ET) = 100 - 85 = 15\]
So 15 employees read both.
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In a company with 100 employees, a survey finds that 45 read Business Today, 55 read Economic Times, and 10 read neither. How many employees read both magazines?
Answer: B
Let BT be the set of employees who read Business Today and ET be the set for Economic Times.
Total employees = 100.
Number of employees who read neither magazine = 10.
Therefore, the number of employees who read at least one magazine is n(BT ∪ ET) = 100 - 10 = 90.
Given: n(BT) = 45, n(ET) = 55.
Using the inclusion-exclusion principle:
n(BT ∪ ET) = n(BT) + n(ET) - n(BT ∩ ET)
90 = 45 + 55 - n(BT ∩ ET)
90 = 100 - n(BT ∩ ET)
n(BT ∩ ET) = 100 - 90 = 10.
So, 10 employees read both magazines.
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What is the cardinality of the set of all vowels in the English alphabet?
Answer: C
The vowels are a, e, i, o and u.
\[ n = 5 \]
Cardinality is simply the count of elements.
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If A = {1, 2} and B = {1, 2, 3}, then A - B is:
Answer: B
The set difference A - B contains all elements that are in set A but not in set B.
A = {1, 2}
B = {1, 2, 3}
All elements of A (1 and 2) are also present in B. Therefore, there are no elements that are in A but not in B.
A - B = ∅ (the empty set).
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A group of 75 students went to an amusement park. 25 of them went on the Ferris Wheel, 35 went on the Roller Coaster, and 10 went on both. How many students went on neither ride?
Answer: C
Let F be the set for Ferris Wheel and R for Roller Coaster.
n(F) = 25, n(R) = 35, n(F ∩ R) = 10.
Number of students who went on at least one ride = n(F ∪ R) = n(F) + n(R) - n(F ∩ R) = 25 + 35 - 10 = 50.
Total students = 75.
Number of students who went on neither ride = Total - n(F ∪ R) = 75 - 50 = 25.
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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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If n(A) = 30 and n(B) = 40, and the two sets are disjoint, what is n(A Δ B)?
Answer: D
The symmetric difference A Δ B = (A ∪ B) - (A ∩ B).
Since the sets are disjoint, n(A ∩ B) = 0.
For disjoint sets, n(A ∪ B) = n(A) + n(B) = 30 + 40 = 70.
Therefore, n(A Δ B) = n(A ∪ B) - 0 = 70.
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