Questions covered in Hard

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. If $A$ and $B$ are two sets, then $A \cap (A \cup B)'$ is equal to
2. Let $A = \{1, 2, 3, \dots, 10\}$. The number of subsets of $A$ containing 1 and 2 but not 3 is
3. If $X = \{4^n - 3n - 1 : n \in N\}$ and $Y = \{9(n-1) : n \in N\}$, then
4. If $A \Delta B = A \cup B$, then
5. Two finite sets have $m$ and $n$ elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of $m$ and $n$ are
6. Prove that $A \subseteq B \iff B' \subseteq A'$.
7. If $P(A) = P(B)$, show that $A = B$.
8. Let $A, B, C$ be sets. Prove that $(A - B) \cap (A - C) = A - (B \cup C)$.
9. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis? (Assume everyone likes at least one).
10. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find the number of people who read exactly one newspaper.
11. A college awarded 38 medals in football, 15 in basketball and 20 in cricket. If these medals went to a total of 58 men and only 3 men got medals in all the three sports, how many received medals in exactly two of the three sports?
12. Case Study: In a survey of 25 students, it was found that 15 had taken Mathematics, 12 had taken Physics and 11 had taken Chemistry, 5 had taken Mathematics and Chemistry, 9 had taken Mathematics and Physics, 4 had taken Physics and Chemistry and 3 had taken all three subjects.
    (i) Find the number of students who had taken only Physics.
    (ii) Find the number of students who had taken only Mathematics.
    (iii) Find the number of students who had taken at least one of the subjects.