Sets: Mixed Practice

Before you begin#

Twelve ques­tions on sets, from nota­tion to count­ing. For every count­ing ques­tion, draw a Venn dia­gram first and fill it from the inner­most region out­wards. It takes a minute and saves you from count­ing the over­laps twice.

Ques­tions#

1. Write $\{x : x \in \mathbf{Z}, x^2 < 10\}$ in ros­ter form and $\{5, 10, 15, 20, 25\}$ in set-builder form.
2. How many sub­sets and how many proper sub­sets does $\{p, q, r, s\}$ have?
3. Is $\{x \in \mathbf{R} : x^2 + 1 = 0\}$ empty? Is $\{x \in \mathbf{Z} : 2x = 7\}$?
4. Write $\{x \in \mathbf{R} : -1 < x \le 4\}$ as an inter­val and draw it on a num­ber line.
5. $U = \{1, 2, \ldots, 9\}$, $A = \{1, 2, 3, 4, 5\}$, $B = \{4, 5, 6, 7\}$. Find $A \cup B$, $A \cap B$, $A - B$, $A'$, $(A \cap B)'$, $A' \cup B'$.
6. Ver­ify De Mor­gan's sec­ond law using ques­tion 5.
7. Show that $A \cap (A \cup B) = A$ for all sets $A, B$.
8. If $A \subset B$, what are $A \cup B$ and $A \cap B$? Prove your answer.
9. $n(U) = 70$, $n(A) = 40$, $n(B) = 25$, $n(A \cap B) = 12$. Find $n(A' \cap B')$.
10. In a group of $90$ peo­ple, $55$ speak Tel­ugu and $48$ speak Hindi, and every­one speaks at least one of the two. How many speak both? How many speak only Hindi?
11. A col­lege has $300$ stu­dents. Of these, $150$ study math­e­mat­ics, $120$ physics and $100$ chem­istry; $50$ take maths and physics, $40$ physics and chem­istry, $45$ maths and chem­istry, and $20$ take all three. How many study none of the three? How many study exactly one?
12. Find sets $A$, $B$, $C$ with $A \cap B = A \cap C$ but $B \ne C$. (So you can­not "can­cel" a set from an inter­sec­tion.)

Answers#