Multiple Choice

1. Let $A = \{2, 4, 6, 8\}$ and $B = \{3, 4, 5, 6\}$. Find the union of the sets, $A \cup B$.

   $\{4, 6\}$

   $\{2, 3, 4, 5, 6, 8\}$

   $\{2, 3, 4, 5, 6\}$

   $\{2, 3, 5, 8\}$

2. Given sets $P = \{a, b, c, d, e\}$ and $Q = \{c, d, f, g\}$, determine the intersection, $P \cap Q$.

   $\{c, d\}$

   $\{a, b, c, d, e, f, g\}$

   $\{a, b, e, f, g\}$

   $\{c\}$

3. If $U = \{1, 2, 3, 4, 5, 6\}$ is the universal set, and $A = \{1, 3, 5\}$, what is the complement of A, denoted as $A'$?

   $\{2, 4, 6\}$

   $\emptyset$

   $\{1, 3, 5\}$

   $\{2, 4\}$

4. Sets $M$ and $N$ are defined as $M = \{x | x \text{ is an even integer}\}$ and $N = \{x | x \text{ is an odd integer}\}$. What is $M \cap N$?

   $\emptyset$

   $N$

   $\{0\}$

   $M$

5. Consider the sets $A = \{10, 11, 12, 13\}$ and $B = \{12, 13, 14, 15\}$. Calculate the set difference $A - B$.

   $\{14, 15\}$

   $\{10, 11, 14, 15\}$

   $\{12, 13\}$

   $\{10, 11\}$

6. The set difference $A - B$ can be expressed in terms of intersection and complement using which equivalent form?

   $A \cap B'$

   $A \cup B'$

   $A' \cap B$

   $A \cap B$

7. Two sets, $A$ and $B$, are described as disjoint sets. What defining characteristic must their intersection satisfy?

   $A \cap B = \emptyset$

   $A \cap B$ must be equal to $A \cup B$

   $A \cap B$ must contain at least one element

   $A \cap B$ must equal $A$

8. If $A$ is a subset of $B$ (i.e., $A \subset B$), what is the resulting set for the intersection $A \cap B$?

   $B$

   $A$

   $A \cup B$

   $\emptyset$

9. If $n(A) = 25$, $n(B) = 30$, and $n(A \cap B) = 10$, what is the cardinality of the union, $n(A \cup B)$?

   55

   65

   45

   40

10. In a universal set $U$, if $n(U) = 100$ and $n(A) = 75$, what is the number of elements in the complement of $A$, $n(A')$?

    25

    175

    35

    75

11. If $n(A) = 15$, $n(B) = 20$, and $n(A \cup B) = 30$, find the number of elements in the intersection, $n(A \cap B)$.

    35

    5

    10

    15

12. Which operation correctly defines the Symmetric Difference of sets $A$ and $B$, denoted $A \Delta B$?

    $A \cap B'$

    $A \cup B$

    $(A \cup B) \cap (A \cap B)$

    $(A - B) \cup (B - A)$

13. Given $A = \{1, 2, 3, 4, 5\}$ and $B = \{3, 5, 7, 9\}$, calculate the Symmetric Difference, $A \Delta B$.

    $\{3, 5\}$

    $\{1, 2, 4, 7, 9\}$

    $\{7, 9\}$

    $\{1, 2, 4\}$

14. If $n(A) = 12$ and $n(A \cap B) = 4$, determine the cardinality of the set difference $n(A - B)$.

    16

    8

    12

    4

15. The set $A = \{x | x \text{ is a prime number less than } 10\}$ and $B = \{1, 3, 5, 7\}$. Find $(A \cup B)$. (Note: Prime numbers are greater than 1).

    $\{1, 2, 5, 7\}$

    $\{2, 3, 5, 7\}$

    $\{1, 3, 5, 7\}$

    $\{1, 2, 3, 5, 7\}$

16. In a class of 40 students, 25 study Math (M) and 20 study Science (S). If 5 students study neither subject, how many students study both Math and Science?

    10

    15

    20

    5

17. Which of the following formulas represents the Principle of Inclusion-Exclusion for finding $n(A \cup B)$?

    $n(A) + n(B)$

    $n(A) + n(B) + n(A \cap B)$

    $n(A) - n(B) + n(A \cap B)$

    $n(A) + n(B) - n(A \cap B)$

18. Using De Morgan's Law, the complement of the union of two sets, $(A \cup B)'$, is equivalent to:

    $A' \cup B'$

    $A \cup B$

    $A' \cap B'$

    $(A \cap B)$

19. Given $U = \{a, b, c, d, e\}$, $A = \{a, b\}$, $B = \{b, c, d\}$. Find $(A \cap B)'$.

    $\{c, d, e\}$

    $\{a, b, c, d\}$

    $\{b\}$

    $\{a, c, d, e\}$

20. If $A$ and $B$ are overlapping sets, which of the following statements must be true?

    $A \cap B = \emptyset$

    $A \cap B \neq \emptyset$

    $A \cup B = U$

    $A$ is a subset of $B$

21. Simplify the set expression $(A - B) \cup (A \cap B)$.

    $A'$

    $A \cap B'$

    $A$

    $B$

22. In a class of 60 students, 42 study Math ($M$) and 35 study Science ($S$). If exactly 15 students study neither Math nor Science, how many students study both subjects?

    32

    22

    30

    25

23. Sets $P$ and $Q$ are disjoint, with $n(P) = 18$ and $n(Q) = 25$. If $R = P \Delta Q$ (symmetric difference), what is $n(R)$?

    36

    7

    43

    50

24. Given non-empty sets $X$ and $Y$, simplify the expression $X' \cap (Y \cup X)$.

    $Y - X$

    $X \cup Y$

    $Y'$

    $X'$

25. Set $A$ has $n(A) = 55$ elements and Set $B$ has $n(B) = 45$ elements. If the universal set $U$ contains 80 elements, what is the minimum possible number of elements in $A \cap B$?

    10

    45

    20

    0

26. If $U = \{x | x \text{ is an integer, } 1 \le x \le 10\}$ and $A = \{1, 3, 5, 7, 9\}$. Which statement is FALSE regarding $A'$?

    $A \cap A'$ is the empty set.

    $A'$ contains an even number of elements.

    $A' = U - A$

    $A \cup A' = A$

27. Let $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5, 6\}$, and $C = \{5, 6, 7, 8\}$. Evaluate the set $(A \Delta B) \cap C$.

    $\{3, 4\}$

    $\{5, 6\}$

    $\{5\}$

    $\{1, 2, 3, 4, 5, 6\}$

28. Given $n(A - B) = 30$, $n(B - A) = 25$, and $n(A \cap B)$ is exactly $1/4$ of $n(A)$. Calculate $n(A \cup B)$.

    85

    60

    65

    75

29. Which of the following set expressions is always equivalent to $(A \cup B) - A$?

    $B \cup A'$

    $(A' \cap B)'$

    $B \cap A'$

    $A \cap B'$

30. A group of 100 musicians was surveyed. 68 play the violin ($V$) and 54 play the piano ($P$). If all 100 musicians play at least one instrument, how many musicians play exactly one instrument?

    22

    50

    78

    80