Chapter Five: Sets – Additional Mathematics Form Two
Different objects with common characteristics can be categorized and well-studied when arranged together in groups. For instance, a group of form two students’ names, a collection of coins, a group of even numbers less than ten, and so on.
In this chapter, you will learn about operations on sets and number of elements in a set. The competencies developed in this chapter will help you to fulfill various tasks such as organizing, creating, categorizing objects, among many other applications.
The Concept of Sets
A set is a collection of well-defined distinct objects. It is denoted by a capital letter, with its elements enclosed within a curled brackets { } and separated by commas. The elements of a set are represented by small letters, numbers, symbols or words.
Set Notation
If a is an element of set A, then in set notation is written as a ∈ A.
If a is not the element of set A, symbolically it is written as a ∉ A.
A set containing all elements under consideration is called a universal set, it is denoted by the letter U.
Example
If U = {2, 7, 8, 9, 12, 13, 15, 16, 17, 19, 21, 29}, A = {2, 8, 13, 17, 29}, B = {7, 12, 13, 16, 17, 21}, and C = {8, 9, 12, 16, 17, 19}, find:
(a) A ∪ B ∪ C
(b) A ∩ B ∩ C
(c) B’ ∪ A
Solution
(a) A ∪ B = {2, 8, 13, 17, 29} ∪ {7, 12, 13, 16, 17, 21} = {2, 7, 8, 12, 13, 16, 17, 21, 29}
A ∪ B ∪ C = {2, 7, 8, 12, 13, 16, 17, 21, 29} ∪ {8, 9, 12, 16, 17, 19} = {2, 7, 8, 9, 12, 13, 16, 17, 19, 21, 29}
(b) A ∩ B = {2, 8, 13, 17, 29} ∩ {7, 12, 13, 16, 17, 21} = {13, 17}
A ∩ B ∩ C = {13, 17} ∩ {8, 9, 12, 16, 17, 19} = {17}
(c) B’ = {2, 8, 9, 15, 19, 29}
B’ ∪ A = {2, 8, 9, 15, 19, 29} ∪ {2, 8, 13, 17, 29} = {2, 8, 9, 13, 15, 17, 19, 29}
Activity 5.1: Collecting together items of the same characteristics
Individually or in groups, perform the following tasks:
- Using manila card or any other card of your choice, prepare one hundred pieces of cards with convenient dimensions.
- Using marker pens or pencils, label the pieces of cards prepared in task 1 with numbers from 1 to 20 in five pairs.
- From task 2, prepare a set A whose elements are even numbers less than 10 and odd numbers greater than 10.
- From task 2, prepare a set B of odd numbers less than 17.
- From task 2, prepare a set C of prime numbers.
- From task 2, form a set D whose elements are either in sets A, B, or C without repeating the elements.
- From task 2, form a set E of common elements found in sets A, B, and C without repeating the elements.
- List the elements that do not belong to a set B.
- Suggest the names of the sets operations used in tasks 6, 7, and 8.
- Identify various sets that you have encountered in your daily life.
- Are there any sets identified in task 10 with common elements? If so list them.
- Share your results with other students through discussions for more inputs.
Operations on Sets
Union of Sets
The union of two or more sets is the set that contains all the elements that are in all the sets without repetition. The union is denoted by the cup symbol “∪”.
If x ∈ (A ∪ B ∪ C), then x ∈ A, x ∈ B or x ∈ C.
Intersection of Sets
The intersection of two or more sets is the set that contains all the elements that are common to all the sets. The intersection is denoted by the cap symbol “∩”.
If x ∈ (A ∩ B ∩ C), then x ∈ A, x ∈ B, and x ∈ C.
Complement of a Set
The complement of a set A is the set that contains all the elements that are in the universal set but not in the set A. It is denoted by A’ or Ac.
If x ∈ A’, then x ∉ A.
Venn Diagram Representation
Venn diagrams are used to show the logical relationship between sets.
Number of Elements in a Set
Cardinality of Sets
The number of elements of a set is a count of individual elements in a set. It is also known as the cardinality of the set and is denoted by n(A), reads “the number of elements in the set A”.
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(A ∩ C) – n(B ∩ C) + n(A ∩ B ∩ C)
Example
If n(U) = 68, n(A) = 41, n(C) = 21, n(B) = 17, n(A ∩ B) = 19, n(A ∩ C) = 23, n(B ∩ C) = 11, and n(A ∩ B ∩ C) = 27, find:
(a) n(A ∪ B ∪ C)
(b) n(A ∪ B ∪ C)’
Solution
(a) Using the formula:
n(A ∪ B ∪ C) = 41 + 17 + 21 – 19 – 23 – 11 + 27
= 106 – 53 = 53
(b) n(A ∪ B ∪ C)’ = n(U) – n(A ∪ B ∪ C)
= 68 – 53 = 15
Example
A secondary school carried out a monthly test and the results were as follow. Out of 1,350 candidates, 600 passed Geography, 700 passed History, 350 passed English Language and 50 failed all the three subjects. Also, 200 passed Geography and History, 150 passed Geography and English Language, 100 passed History and English Language.
(a) How many candidates passed in all the three subjects?
(b) Illustrate the given information in a Venn diagram.
Solution
Let G, H, and E represent sets of candidates who passed Geography, History, and English respectively.
n(U) = 1350, n(G) = 600, n(H) = 700, n(E) = 350
n(G ∪ H ∪ E)’ = 50, n(G ∩ H) = 200, n(G ∩ E) = 150, n(H ∩ E) = 100
n(G ∪ H ∪ E) = 1350 – 50 = 1300
Using the formula:
1300 = 600 + 700 + 350 – 200 – 150 – 100 + n(G ∩ H ∩ E)
1300 = 1200 + n(G ∩ H ∩ E)
n(G ∩ H ∩ E) = 100
Therefore, 100 candidates passed all three subjects.
Practical Applications
Real-life Set Applications
Sets are used in various real-life situations:
- Organizing student data in schools
- Categorizing products in supermarkets
- Classifying books in libraries
- Grouping employees in organizations
- Sorting data in computer systems
Example
An investigator was paid 1,000 Tanzanian shillings per person interviewed about like and dislike on food served for lunch. He reported that 200 like rice, 190 like chips, 260 like Ugali, 60 like chips and rice, 40 like chips and ugali, 100 like rice and ugali, 30 like all three types of food, while 105 people do not like any food at all.
(a) Summarize this information in a Venn diagram.
(b) Hence, calculate the amount of money paid to the investigator.
Solution
Let R, C, and M represent sets of people who like rice, chips and ugali respectively.
Using Venn diagram principles and the inclusion-exclusion principle, we can calculate the total number of people interviewed.
Total people = Sum of all regions in the Venn diagram
Amount paid = Total people × 1000 Tsh
Exercise 1
- Given that U = {a, b, c, d, f, g, h, j, k, w, l, m, n, t}, A = {a, d, f, g, h, j, k}, B = {g, h, j, k, l, m, n}, and C = {a, b, c, d, f, g, h, j}
- Present the given sets in a Venn diagram.
- Shade the region represented by (C’ ∩ A’) ∪ (B’ ∪ C)’.
- Shade the region not represented by (A ∪ B’) ∩ (B ∪ C’).
- If U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 3, 4} and B = {5, 6, 7}
- Find: (i) A’ ∪ B’ (ii) A’ ∩ B’
- Show that (A ∪ B)’ = A’ ∩ B’.
Exercise 2
- Given three sets, A, B, and C such that n(A) = 70, n(B) = 62, n(C) = 59, n(A ∩ B) = 25, n(A ∩ C) = 15, n(B ∩ C) = 19, and n(A ∩ B ∩ C) = 12. Find:
- n(A ∪ B ∪ C)
- n(A ∩ B’)
- n(B ∩ C’)
- n(A’ ∩ C)
- Each student in a Form two class of 40 students plays at least one of the following games; volleyball, football, and tennis. There are 18 students who play volleyball, 20 students who play football, and 27 students who play tennis. If 7 students play volleyball and football, 12 students play football and tennis, and 4 students play all games. Use formulas to calculate the number of students who play:
- Volleyball and tennis.
- Volleyball and tennis but not football.
- Just one game.
- Football or tennis.
Chapter Summary
- Set is a well-defined collection of distinct objects.
- Union of sets is a set that contains all elements of all sets.
- Intersection of sets is a set that contains all elements that are common in all sets.
- Complement of a set is the set that contain all elements of the universal set that are not in the given set.
- Universal set is a set which contains all the elements under consideration.
- Venn diagram is the diagram used to show the logical relationship between sets.
- Number of elements in a set is the simple count of individual elements in a set.
- The formula for three sets: n(A∪B∪C) = n(A) + n(B) + n(C) – n(A∩B) – n(A∩C) – n(B∩C) + n(A∩B∩C)
Revision Exercise
- If A, B, and C are subsets of the universal set U, then shade the region represented by each of the following:
- (A ∩ B’) ∪ (A ∪ B)’
- A ∩ (B ∪ C)’
- (A ∩ C) ∪ (C’ ∪ A)’
- If U = {105, 115, 125, 135, 145, 155, 165, 175, 185}, D = {115, 125, 135, 145, 155}, E = {105, 115, 145, 165}, and F = {115, 145, 165, 175}, then find:
- (D ∪ E’) ∩ (E ∪ F’)
- (D ∪ E ∪ F)’
- (D ∩ E) ∪ (E ∩ F)
- Given that U = {red, black, pink, yellow, blue, white, magenta, purple}, A = {red, black, pink}, B = {black, yellow, blue}, and C = {black, white, magenta} Find:
- (A’ ∪ B’)’ ∪ (A’ ∪ B)’
- A’ ∪ (C ∩ B)’
- B ∪ (A ∩ C)