Faculty of Engineering & Technology, CVM University
A.Y.2024-25: EVEN SEMESTER
202040405: DISCRETE MATHEMATICS
Tutorial 1: Set Theory
1. List the members of these sets.
( a ) {𝑥 | 𝑥 is a real number such that 𝑥 2 = 1 }
( b ) {𝑥 | 𝑥 is a positive integer less than 12 }
( c ) {𝑥 | 𝑥 is the square of an integer and 𝑥 < 100 }
(d ) {𝑥 | 𝑥 is an integer such that 𝑥 2 = 2 }
2. Use set builder notation to give a description of each of these sets.
(𝑎) {0, 3, 6, 9,12} (𝑏) {−3, −2, −1, 0, 1, 2, 3}
3. For each of the following sets, determine whether 2 is an element of that set.
( 𝑎 ) {𝑥 ∈ 𝑹 | 𝑥 is an integer greater than 1 }
( 𝑏 ) {𝑥 ∈ 𝑹 | 𝑥 is the square of an integer }
(𝑐) {2, {2}} (𝑑) {{2}, {{2}}} (𝑒) {{2}, {2, {2}}} (𝑓) {{{2}}}
Also check whether {2} is an element of which of the above sets.
4. Determine whether each of these statements is true or false.
(𝑎) 0 ∈ 𝜑 (𝑏) 𝜑 ⊂ {0} (𝑐) {𝜑} ⊆ {𝜑} (𝑑) {0} ∈ {0}
(𝑒) 𝜑 ∈ {𝜑} (𝑓) {𝜑} ∈ {{𝜑}} (𝑔) {{𝜑}} ⊂ {𝜑, {𝜑}}
5. Use a Venn diagram to illustrate the following relationships:
(𝑎) 𝐴 ⊆ 𝐵 𝑎𝑛𝑑 𝐵 ⊆ 𝐶 (𝑏) 𝐴 ⊂ 𝐵 𝑎𝑛𝑑 𝐵 ⊂ 𝐶 (𝑐) 𝐴 ⊂ 𝐵 𝑎𝑛𝑑 𝐴 ⊂ 𝐶.
6. Find two sets 𝐴 and 𝐵 such that 𝐴 ∈ 𝐵 and 𝐴 ⊆ 𝐵.
7. What is the cardinality of each of these sets?
(𝑎) 𝜑 (𝑏) {𝜑} (𝑐) {{𝑎}} (𝑑) {𝜑, {𝜑}} (𝑒) {𝜑, {𝜑}, {𝜑, {𝜑}}}.
8. a) Find the power set of {𝜑, {𝜑}}.
b) Let A = {a, b, c, d} and B = {y, z}.
Find i) A × B ii) B × A
9. How many elements does each of these sets have where 𝑎 and 𝑏 are distinct elements?
(𝑎) 𝑃({𝑎, 𝑏, {𝑎, 𝑏}}) (𝑏) 𝑃 ({𝜑, 𝑎, {𝑎}, {{𝑎}}}) (𝑐) 𝑃(𝑃(𝜑)).
10. Determine whether each of these sets is the power set of a set, where 𝑎 and 𝑏 are distinct
elements.
(𝑎) 𝜑 (𝑏) {𝜑, {𝑎}} (𝑐) {𝜑, {𝑎}, {𝜑, {𝑎}, {𝜑, 𝑎}}} (𝑑) {𝜑, {𝑎}, {𝑏}, {𝑎, 𝑏}}
11. (i) Find 𝐴2 if a) A = {0, 1, 3}, b) A = {1, 2, a, b}.
(ii) Find A3 if a) A = {a}, b) A = {0, a}.
12. How many different elements does A × B have if A has m elements and B has n elements?
13. Draw the Venn diagrams for each of these combinations:
(𝑎) 𝐴̅⋂𝐵̅ ⋂𝐶̅ (𝑏) (𝐴 − 𝐵)⋃(𝐴 − 𝐶)⋃(𝐵 − 𝐶) (𝑐) (𝐴⋂𝐵̅ )⋃(𝐴⋂𝐶̅ )
(𝑑) (𝐴⋂𝐵)⋃(𝐶⋂𝐷) (𝑒) 𝐴 − (𝐵⋂𝐶⋂𝐷)
Use separate sheet of paper to answer this question.
14. In a recent survey, people were asked if they took a vacation in the summer, winter or
spring in the last year. The results were: 73 took a vacation in the summer, 51 took a
vacation in the winter, 27 took a vacation in the spring, and 2 had taken no vacation. Also,
10 had taken vacations at all three times, 33 had taken both a summer and a winter
vacation, 18 had taken only a winter vacation and 5 had taken both a summer and a spring
but not a winter vacation.
( a ) How many people had been surveyed?
( b ) How many people had taken vacations at exactly two times of the year?
( c ) How many people had taken vacations during at most one time of the year?
( d ) What percentage had taken vacations during both summer and winter but not spring?
𝑨𝒏𝒔: (𝒂) 𝟏𝟎𝟓 (𝒃) 𝟐𝟖 (𝒄) 𝟔𝟕 (𝒅) 𝟐𝟏. 𝟗𝟎𝟒𝟖 %
15. Find the sets 𝐴 and 𝐵 if 𝐴 − 𝐵 = {1, 5, 7, 8}, 𝐵 − 𝐴 = {2, 10}, and 𝐴⋂𝐵 = {3, 6, 9}.
16. What can you say about the sets 𝐴 and 𝐵 if we know that
(𝑎) 𝐴⋃𝐵 = 𝐴 (𝑏) 𝐴⋂𝐵 = 𝐴 (𝑐) 𝐴 − 𝐵 = 𝐴
(𝑑) 𝐴⋂𝐵 = 𝐵⋂𝐴 (𝑒) 𝐴 − 𝐵 = 𝐵 − 𝐴
17. Show that if 𝐴 is a subset of a universal set 𝑈, then
(𝑎) 𝐴 ⊕ 𝐴 = 𝜑 (𝑏) 𝐴 ⊕ 𝜑 = 𝐴 (𝑐) 𝐴 ⊕ 𝑈 = 𝐴̅ (𝑑) 𝐴 ⊕ 𝐴̅ = 𝑈.
18. If 𝐴, 𝐵 and 𝐶 are sets such that 𝐴 ⊕ 𝐶 = 𝐵 ⊕ 𝐶, can we conclude that 𝐴 = 𝐵?
19. Suppose that the universal set is 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Express the sets
(𝑎) {3, 4, 5} (𝑏) {1, 3, 6, 10} (𝑐) {2, 3, 4, 7, 8, 9}
with bit strings where the 𝑖th bit in the string is 1 if 𝑖 is in the set and 0 otherwise.
Also, find the set specified by each of the bit strings
(𝑎) 11 1100 1111 (𝑏) 01 0111 1000 (𝑐) 10 0000 0001
20. What subsets of a finite universal set do these bit strings represent?
( a ) the string with all zeros ( b ) the string with all ones
