If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is A ∪ B?
Answer: B
The union collects every element of either set, listed once.
\[ A \cup B = \{1, 2, 3, 4, 5, 6\} \]
3 and 4 belong to both but are not repeated.
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If A = {a, b, c, d} and B = {c, d, e, f}, what is A ∩ B?
Answer: C
The intersection keeps only the elements common to both sets.
\[ A \cap B = \{c, d\} \]
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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 contains everything in the universal set that is not in \( A \).
\[ A' = \{2, 4, 6, 8\} \]
Together \( A \) and \( A' \) make up the whole of \( U \) and share nothing.
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If a set A has 5 elements, what is the number of elements in its power set, n(P(A))?
Answer: D
The power set contains every subset, and each element is either in or out.
\[ n(P(A)) = 2^{n(A)} = 2^{5} = 32 \]
That count includes the empty set and \( A \) itself.
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If n(A) = 20, n(B) = 30 and n(A ∪ B) = 40, find n(A ∩ B).
Answer: B
We use the formula for the cardinality of the union of two sets:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
Substituting the given values:
40 = 20 + 30 - n(A ∩ B)
40 = 50 - n(A ∩ B)
n(A ∩ B) = 50 - 40 = 10
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In a class of 50 students, 30 like Math, 25 like Science, and 10 like both. How many students like neither Math nor Science?
Answer: A
Let M be the set of students who like Math and S be the set of students who like Science.
n(M) = 30, n(S) = 25, n(M ∩ S) = 10.
Number of students who like at least one subject is n(M ∪ S) = n(M) + n(S) - n(M ∩ S) = 30 + 25 - 10 = 45.
Total students = 50.
Number of students who like neither = Total students - n(M ∪ S) = 50 - 45 = 5.
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Which of the following represents the shaded region in the Venn diagram?
Answer: C
The shaded region lies inside circle A but outside circle B - the part of \( A \) that \( B \) does not overlap.
\[ A - B = \{\, x \mid x \in A \text{ and } x \notin B \,\} \]
\( A \cap B \) would be the lens where they overlap, and \( B - A \) the mirror image on the other side.
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If A = {x : x is a prime number less than 10}, find the cardinality of A.
Answer: B
We need to list the prime numbers less than 10.
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.
The prime numbers less than 10 are 2, 3, 5, and 7.
So, A = {2, 3, 5, 7}.
The cardinality of A, denoted n(A), is the number of elements in the set, which is 4.
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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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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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