By the end of this chapter, you will be able to:
Mastering these skills will help you analyze and solve practical business challenges with clarity and precision.
Compiled by: Samuel Maina
Business professionals across all sectors in Kenya encounter situations involving collections of items, people, or data sets. Mastering set theory allows them to organize, analyze, and interpret such collections effectively, facilitating decision-making processes in environments ranging from banks and SACCOs to hospitals and county offices. This chapter introduces the fundamental concepts of set theory, equipping students with the tools to solve problems involving unions, intersections, and complements of sets in practical professional contexts.
Set theory forms the foundation for understanding collections of objects or elements in mathematics and real-world applications. A set is a well-defined collection of distinct objects considered as an entity. In professional fields, set theory helps organize data such as client lists, inventory items, or project tasks by grouping related elements and analyzing their relationships. The notation and operations of set theory enable efficient problem-solving and data interpretation.
Sets are collections of distinct elements, typically denoted by capital letters such as \(A\), \(B\), or \(C\). Elements of a set are enclosed in curly braces, for example, \(A = \{1, 2, 3, 4\}\). The membership of an element \(x\) in a set \(A\) is indicated by \(x \in A\), while \(x otin A\) means \(x\) is not in \(A\). Sets can be finite or infinite, and elements can be numbers, objects, or even other sets.
The fundamental operations on sets include:
Union (\(A \cup B\)): the set containing all elements in \(A\) or \(B\),
Intersection (\(A \cap B\)): the set containing elements common to both \(A\) and \(B\),
$$ A = \{a_1, a_2, ..., a_n\} $$
$$ x \in A \quad \text{means} \quad x \text{ is an element of } A $$
Example 1: A county referral hospital has a set \(P\) of patients admitted on Monday: \(P = \{101, 102, 103, 104\}\). Is patient 105 in the set?
Given: \(P = \{101, 102, 103, 104\}\), check if \(105 \in P\).
Since \(105\) is not listed,
Answer: \(105 otin P\)
Example 2: A retail business has a set \(I\) of items in stock: \(I = \{\text{soap}, \text{shampoo}, \text{toothpaste}\}\). Write the set notation for the items.
Given: Items soap, shampoo, toothpaste.
$$ I = \{\text{soap}, \text{shampoo}, \text{toothpaste}\} $$
Answer: \(I = \{\text{soap}, \text{shampoo}, \text{toothpaste}\}\)
Example 3: Define the universal set \(U\) as the first 10 natural numbers. Write set \(A = \{2, 4, 6, 8, 10\}\) and find \(x \in A\) and \(x otin A\).
Given: \(U = \{1,2,3,4,5,6,7,8,9,10\}\), \(A = \{2,4,6,8,10\}\).
Check if \(5 \in A\):
\(5\) is not in \(A\), so \(5 otin A\).
Answer: \(5 otin A\)
Example 4: A SACCO client list \(S = \{C1, C2, C3\}\). Check if \(C2\) is a member.
Given: \(S = \{C1, C2, C3\}\).
Since \(C2\) is in the set,
Answer: \(C2 \in S\)
Example 5: A school club has members \(M = \{John, Mary, Peter\}\). Is "Jane" a member?
Given: \(M = \{John, Mary, Peter\}\).
"Jane" is not listed,
Answer: "Jane"
otin M
The union of two sets \(A\) and \(B\), denoted \(A \cup B\), is the set of all elements that belong to \(A\) or \(B\) or both. The intersection, \(A \cap B\), is the set of elements common to both \(A\) and \(B\). These operations allow professionals to combine or compare different groups of data effectively.
The formulas are:
$$ A \cup B = \{x : x \in A \text{ or } x \in B\} $$
$$ A \cap B = \{x : x \in A \text{ and } x \in B\} $$
Example 1: At a university, set \(A\) represents students enrolled in a math course: \(A = \{1, 2, 3, 4\}\), and set \(B\) in a computer course: \(B = \{3, 4, 5, 6\}\). Find \(A \cup B\) and \(A \cap B\).
Given: \(A = \{1, 2, 3, 4\}\), \(B = \{3, 4, 5, 6\}\).
$$ A \cup B = \{1, 2, 3, 4, 5, 6\} $$
$$ A \cap B = \{3, 4\} $$
Answer:
\(A \cup B = \{1, 2, 3, 4, 5, 6\}\)
\(A \cap B = \{3, 4\}\)
Example 2: A retail business has sets \(C = \{\text{soap}, \text{toothpaste}\}\) and \(D = \{\text{toothpaste}, \text{shampoo}\}\). Find \(C \cup D\) and \(C \cap D\).
Given: \(C = \{\text{soap}, \text{toothpaste}\}\), \(D = \{\text{toothpaste}, \text{shampoo}\}\).
$$ C \cup D = \{\text{soap}, \text{toothpaste}, \text{shampoo}\} $$
$$ C \cap D = \{\text{toothpaste}\} $$
Answer:
\(C \cup D = \{\text{soap}, \text{toothpaste}, \text{shampoo}\}\)
\(C \cap D = \{\text{toothpaste}\}\)
Example 3: At a county government office, set \(E = \{\text{tax officers}, \text{auditors}\}\), set \(F = \{\text{auditors}, \text{clerks}\}\). Find \(E \cup F\) and \(E \cap F\).
Given: \(E = \{\text{tax officers}, \text{auditors}\}\), \(F = \{\text{auditors}, \text{clerks}\}\).
$$ E \cup F = \{\text{tax officers}, \text{auditors}, \text{clerks}\} $$
$$ E \cap F = \{\text{auditors}\} $$
Answer:
\(E \cup F = \{\text{tax officers}, \text{auditors}, \text{clerks}\}\)
\(E \cap F = \{\text{auditors}\}\)
Example 4: In a farm cooperative, \(G = \{\text{maize farmers}, \text{beans farmers}\}\), \(H = \{\text{beans farmers}, \text{sorghum farmers}\}\). Find \(G \cup H\) and \(G \cap H\).
Given: \(G = \{\text{maize farmers}, \text{beans farmers}\}\), \(H = \{\text{beans farmers}, \text{sorghum farmers}\}\).
$$ G \cup H = \{\text{maize farmers}, \text{beans farmers}, \text{sorghum farmers}\} $$
$$ G \cap H = \{\text{beans farmers}\} $$
Answer:
\(G \cup H = \{\text{maize farmers}, \text{beans farmers}, \text{sorghum farmers}\}\)
\(G \cap H = \{\text{beans farmers}\}\)
Example 5: At a hotel, \(J = \{\text{guests using pool}, \text{guests using gym}\} = \{A, B, C\}\), \(K = \{\text{guests using gym}, \text{guests using spa}\} = \{B, C, D\}\). Find \(J \cup K\) and \(J \cap K\).
Given: \(J = \{A, B, C\}\), \(K = \{B, C, D\}\).
$$ J \cup K = \{A, B, C, D\} $$
$$ J \cap K = \{B, C\} $$
Answer:
\(J \cup K = \{A, B, C, D\}\)
\(J \cap K = \{B, C\}\)
The complement of a set \(A\), denoted \(A^{c}\), includes all elements in the universal set \(U\) that are not in \(A\). The difference between two sets \(A\) and \(B\), denoted \(A - B\), contains elements that are in \(A\) but not in \(B\). These operations help isolate exclusive elements and remove overlaps, crucial in data analysis, such as identifying clients who have not completed payments or employees not attending training.
The formulas are:
$$ A^{c} = \{x : x \in U \text{ and } x otin A\} $$
$$ A - B = \{x : x \in A \text{ and } x otin B\} $$
Example 1: At a TVET college, universal set \(U = \{1,2,3,4,5,6,7,8,9,10\}\) represents all students. Let \(A = \{2,4,6,8,10\}\) be those enrolled in a science course. Find \(A^{c}\).
Given: \(U = \{1,2,3,4,5,6,7,8,9,10\}\), \(A = \{2,4,6,8,10\}\).
$$ A^{c} = \{1,3,5,7,9\} $$
Answer:
\(A^{c} = \{1, 3, 5, 7, 9\}\)
Example 2: A bank has clients set \(B = \{\text{John}, \text{Mary}, \text{Peter}, \text{Jane}\}\) and loan defaulters set \(D = \{\text{Mary}, \text{Jane}\}\). Find \(B - D\).
Given: \(B = \{\text{John}, \text{Mary}, \text{Peter}, \text{Jane}\}\), \(D = \{\text{Mary}, \text{Jane}\}\).
$$ B - D = \{\text{John}, \text{Peter}\} $$
Answer:
\(B - D = \{\text{John}, \text{Peter}\}\)
Example 3: At a county government office, universal set \(U = \{\text{officers 1 to 6}\}\), set \(C = \{\text{officers 2, 4, 6}\}\) attended a workshop. Find \(C^{c}\).
Given: \(U = \{1,2,3,4,5,6\}\), \(C = \{2,4,6\}\).
$$ C^{c} = \{1,3,5\} $$
Answer:
\(C^{c} = \{1, 3, 5\}\)
Example 4: In a cooperative, \(F = \{\text{farmers with loans}\} = \{\text{A}, \text{B}, \text{C}, \text{D}\}\), \(R = \{\text{farmers repaid}\} = \{\text{B}, \text{D}\}\). Find \(F - R\).
Given: \(F = \{\text{A}, \text{B}, \text{C}, \text{D}\}\), \(R = \{\text{B}, \text{D}\}\).
$$ F - R = \{\text{A}, \text{C}\} $$
Answer:
\(F - R = \{\text{A}, \text{C}\}\)
Example 5: A retail business universal set \(U = \{\text{all products}\} = \{1,2,3,4,5\}\), set \(P = \{2,3,5\}\) are products in promotion. Find \(P^{c}\).
Given: \(U = \{1,2,3,4,5\}\), \(P = \{2,3,5\}\).
$$ P^{c} = \{1,4\} $$
Answer:
\(P^{c} = \{1, 4\}\)
Venn diagrams visually represent relationships between sets using overlapping circles within a rectangle representing the universal set. They illustrate unions, intersections, complements, and differences, aiding comprehension of set operations and data analysis. In professional contexts, Venn diagrams help identify overlapping client groups, resource allocation, or task distributions.
Venn diagrams follow these principles:
Example 1: In a school, 30 students take English (\(E\)), 20 take Kiswahili (\(K\)), and 10 take both. Draw a Venn diagram and find the number taking either subject.
Given:
Number taking either:
$$ n(E \cup K) = n(E) + n(K) - n(E \cap K) $$
$$ = 30 + 20 - 10 = 40 $$
Answer:
Number taking English or Kiswahili = 40 students
Example 2: At a hospital, 50 staff are nurses (\(N\)), 30 are technicians (\(T\)), 15 are both. Find \(n(N \cup T)\).
Given:
$$ n(N \cup T) = 50 + 30 - 15 = 65 $$
Answer:
Total nurses or technicians = 65 staff
Example 3: A retail store has 100 products. 60 are electronics (\(E\)), 45 are home appliances (\(H\)), 25 are both. Find how many are neither.
Given:
Number in either:
$$ n(E \cup H) = 60 + 45 - 25 = 80 $$
Number neither:
$$ n(U) - n(E \cup H) = 100 - 80 = 20 $$
Answer:
20 products are neither electronics nor home appliances
Example 4: At a cooperative, 70 members grow maize (\(M\)), 50 grow beans (\(B\)), 30 grow both. Total members 100. Find members growing only maize.
Given:
Members growing only maize:
$$ n(M) - n(M \cap B) = 70 - 30 = 40 $$
Answer:
40 members grow only maize
Example 5: At a SACCO, 120 members save regularly (\(S\)), 80 have loans (\(L\)), 50 do both. Find members who save or have loans but not both.
Given:
Members who save or have loans but not both:
$$ n(S \cup L) - n(S \cap L) = (120 + 80 - 50) - 50 = 150 - 50 = 100 $$
Answer:
100 members save or have loans but not both
In a university, set \(A\) has 40 students enrolled in biology, set \(B\) has 30 in chemistry, and 15 are enrolled in both. Find the number of students enrolled in biology or chemistry. (5 marks)
At a retail business, 100 items are stocked. 70 are food items, 60 are beverages, and 40 are both. How many items are neither food nor beverages? (5 marks)
A hospital has 150 staff. 90 are nurses, 60 are doctors, and 40 are both nurses and doctors. Find the number of staff who are either nurses or doctors but not both. (5 marks)
In a cooperative, 100 members grow coffee, 80 grow tea, and 50 grow both. Find the number of members who grow only coffee. (5 marks)
A bank has 200 clients. 120 have savings accounts, 100 have loan accounts, and 70 have both. Find the complement of clients who have both accounts relative to all clients. (5 marks)
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Create a free accountThis chapter introduced the fundamental concepts of set theory, providing a foundation for understanding collections of objects treated as single entities. It explored various types of sets, including the universal set which contains all elements under consideration, the empty or null set with no elements, subsets that are contained within other sets, and distinctions between finite and infinite sets. The chapter then covered key set operations such as unions combining all elements from involved sets, intersections identifying common elements, complements representing elements not in a set, and set difference which highlights elements in one set but not another. Visual representation of these concepts was demonstrated through Venn diagrams, which help illustrate relationships between sets clearly and effectively. Together, these topics equip students with essential tools for analyzing and solving problems involving groups of items in business mathematics and statistics.
In a county referral hospital, there are 120 nurses and 200 doctors. If the set of nurses is \(N\) and the set of doctors is \(D\), find the total number of medical staff assuming no overlap. Calculate \(|N \cup D|\). (2 marks)
A retail business has 150 customers who bought product A and 100 customers who bought product B. If 40 customers bought both products, find the number of customers who bought either product A or B. (3 marks)
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