Question 1: Write the set notation for students who are in set A but not in set B.

Answer: A \ B

Question 2: Express the intersection of sets C and D using set notation.

Answer: C \cap D

Question 3: If x belongs to set E, write the notation for x not belonging to set E.

Answer: x \notin E

Question 4: Simplify the expression: (A ∪ B) \ (A ∩ B).

Answer: (A \ B) \cup (B \ A)

Question 1: Set F contains 20 students, 12 like soccer, and 8 like basketball. If 5 students like both sports, how many students like neither sport? Use set notation to represent your solution.

Answer: Total students = 20; |F ∩ Basketball| = 5; |F ∪ Basketball| = |F| + |Basketball| - |F ∩ Basketball| = 12 + 8 - 5 = 15; Students liking at least one sport = 15; Therefore, students liking neither = 20 - 15 = 5.

Question 2: Given sets G and H where G = {2, 4, 6, 8} and H = {4, 8, 10, 12}, find G \cap H and express it using set notation.

Answer: {4, 8}

Question 1: A survey finds that 30% of people like tea, 50% like coffee, and 15% like both. Represent this information using set notation and Venn diagrams concepts, and find the percentage of people who like neither drink.

Answer: Let T = set of tea drinkers, C = coffee drinkers. |T| = 30%, |C| = 50%, |T \cap C| = 15%. Then, |T \cup C| = 30 + 50 - 15 = 65%. People who like neither = 100% - 65% = 35%.

Question 1: If sets J and K are such that J = {x | x is even} and K = {x | x is a multiple of 4}, describe J \cap K and J \setminus K using set notation.

Answer: J \cap K = {x | x is a multiple of 4}; J \ K = {x | x is even but not a multiple of 4}

Question 2: Prove that the set of all integers divisible by 6 is a subset of the set of integers divisible by 3.

Answer: Any integer divisible by 6 can be written as 6k, which is 3(2k). Since 2k is an integer, every multiple of 6 is also a multiple of 3. Therefore, set of multiples of 6 ⊆ set of multiples of 3.

Question 1: Write the complement of set M, where M = {x | x > 0 and x < 10} within the universal set of integers.

Answer: {x | x ≤ 0 or x ≥ 10}

Question 2: If set N = {1, 3, 5, 7} and set P = {3, 4, 5, 6}, find N \cup P and N \cap P.

Answer: N \cup P = {1, 3, 4, 5, 6, 7}; N \cap P = {3, 5}

Question 1: Mistake: The set of students who like either soccer or basketball is written as S \cap B. Correct this mistake and write the proper notation.

Answer: S \cup B

Question 2: Mistake: Assuming that all elements in set Q are also in set R without checking, leading to Q ⊆ R. Explain whether this is always true or not, and how to verify it.

Answer: It is not always true. To verify Q ⊆ R, check if every element of Q is also in R by listing or comparing sets. If any element of Q is not in R, then Q is not a subset of R.