1. **State the problem:** We are given multiple set theory questions involving membership, types of sets, set notation, and set operations. 2. **Membership statements (True/False):** a. a \in A b. x \in A c. b \in A d. e \in A e. c \in A f. Set A is finite. To determine truth values, we need the definition of set A, which is not provided. So these cannot be answered without more info. 3. **Types of sets:** a. N = \{1,2,3,4,\ldots\} is an infinite set because natural numbers continue indefinitely. b. A = \emptyset is a null set because it contains no elements. c. 5 \in \{3,8,9,14\} is False because 5 is not in the set. d. x = 4 \cdot 3 + 3 \cdot 1 = 12 + 3 = 15 \in \{x : x \text{ is an integer}\} is True because 15 is an integer. 4. **Write sets in roster form:** a. A = \{1,2,3,4,5,6,7,8,9,10\} b. B = \{w,x,y,z\} c. C = \{11,13,15,17,19,21,23\} d. D = \{2,4,6,8,10\} e. E = \{12,14,15,16,18\} (five non-prime numbers > 10) f. F = \{42,43,44,45,46,47,48,49\} g. G = \{2,3,5,7,11,13,17,19,23,29\} h. H = \{April, August\} i. I = \{1,3,5,7,9,11,13,15,17,19\} 5. **Set operations:** Given M = \{1,3,5,7,9,11,13,15,17,19\}, N = \{3,6,9,11,13\} a. M \cap N = \{3,9,11,13\} b. M \cup N = \{1,3,5,6,7,9,11,13,15,17,19\} 6. Given P = natural numbers < 50, Q = natural numbers divisible by 5, R = even numbers 2 to 10: a. P \cap Q = \{5,10,15,20,25,30,35,40,45\} b. P \cap R = \{2,4,6,8,10\} c. Q \cap R = \{10\} d. P \cap Q \cap R = \{10\} e. (P \cup R) \cap Q = Q = \{5,10,15,20,25,30,35,40,45\} f. (P \cap Q) \cup R = \{2,4,5,6,8,10,15,20,25,30,35,40,45\} 7. Given e = \{2,4,6,8,10,12\}, A = \{2,6,10,12\}, B = \{4,8\}, C = \{2,4,8,10\}: a. A' = e \setminus A = \{4,8\} b. B' = e \setminus B = \{2,6,10,12\} c. C' = e \setminus C = \{6,12\} d. A' \cup B' = \{2,4,6,8,10,12\} = e e. B' \cap C' = \{6,12\} f. C' \cap A' = \emptyset g. A \cup B' = \{2,4,6,8,10,12\} = e h. (A \cup B)' = e \setminus (A \cup B) = e \setminus e = \emptyset i. A \cap A' = \emptyset j. A' \cap C = \{4,8\} \cap \{2,4,8,10\} = \{4,8\} 8. Given e = \{1,2,3,\ldots,10\}, A = \{1,2,3,4,5\}, B = \{1,2,6,7\}, C = \{3,5\}: a. n(A \cap B) = n(\{1,2\}) = 2 b. n(B \cap C) = n(\emptyset) = 0 c. n(A \cup B) = n(\{1,2,3,4,5,6,7\}) = 7 d. n((A \cap B)') = n(e \setminus \{1,2\}) = 8 e. n((B \cap C)') = n(e \setminus \emptyset) = 10 f. n((A \cup B)') = n(e \setminus \{1,2,3,4,5,6,7\}) = 3 9. **Venn diagram:** The problem requests a Venn diagram with sets A, B, C showing intersections. This is a conceptual visualization of the above set operations. Final answers are provided above for each question.