By the end of this chapter, you will be able to:
Mastering these skills will help you analyze and organize information clearly, making better decisions in real business situations.
Sets are fundamental in organizing and analyzing data within human resource management, especially when dealing with employee classifications, training groups, and benefits eligibility. Understanding set theory allows HR professionals to efficiently categorize and manipulate groups of employees based on shared characteristics or criteria. This chapter introduces the basic concepts of sets, elements, and the empty set, providing the mathematical foundation necessary for advanced HR data analysis and decision-making.
Basic set theory concepts underpin many HR functions such as grouping employees for training, payroll classifications, or benefits administration. Mastery of these concepts ensures clarity when handling overlapping or exclusive employee categories within organizations like banks, universities, or county government offices.
A set is a well-defined collection of distinct objects, called elements, considered as a single entity. In human resource contexts, a set might represent all employees in a department, all those eligible for a particular benefit, or all staff attending a training program. Sets are usually denoted by capital letters such as \(A\), \(B\), or \(C\).
The formal notation for a set is:
$$ A = \{a_1, a_2, a_3, ..., a_n\} $$
where each \(a_i\) is an element of the set \(A\).
Example 1:
A county government office has a set \(D\) representing employees in the finance department: John, Mary, Peter, and Lucy. Write the set \(D\).
Given: Employees = John, Mary, Peter, Lucy$$ D = \{\text{John}, \text{Mary}, \text{Peter}, \text{Lucy}\} $$ Answer: \(D = \{\text{John}, \text{Mary}, \text{Peter}, \text{Lucy}\}\)
Example 2:
In a SACCO, set \(T\) represents employees attending training on customer service: 12 employees with IDs 101, 102, 103, ..., 112. Express \(T\).
Given: Employee IDs = 101 to 112$$ T = \{101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112\} $$ Answer: \(T = \{101, 102, ..., 112\}\)
Example 3:
A university HR set \(P\) consists of all permanent staff numbered from 201 to 260 inclusive. How many elements are in \(P\)?
Given: Staff numbers 201 to 260 inclusive
Number of elements, \(n = 260 - 201 + 1 = 60\)
Answer: \(n = 60\) elements in set \(P\)
Example 4:
A retail company defines set \(E\) as employees earning above Ksh 50,000 monthly. If there are 45 such employees, express \(E\) in terms of number of elements.
Given: Number of employees \(n = 45\)$$ |E| = 45 $$ Answer: Set \(E\) has 45 elements
Example 5:
If set \(M\) is employees with more than 5 years of service, and set \(N\) is employees with managerial roles, express the union set \(U = M \cup N\) representing employees who are either managers or have more than 5 years’ service.
Given: Sets \(M\) and \(N\)$$ U = M \cup N $$ Answer: \(U\) contains all employees in \(M\) or \(N\) or both
An element is an individual object or member contained within a set. In HR, elements represent individual employees, job titles, or employee IDs. An element either belongs to a set or does not, indicated by the symbols \(\in\) (belongs to) or \( otin\) (does not belong to).
The membership relation is written as:
$$ a \in A \quad \text{if } a \text{ is an element of } A $$
or
$$ a otin A \quad \text{if } a \text{ is not an element of } A $$
Example 1:
Given set \(S = \{\text{Alice}, \text{Bob}, \text{Charles}\}\), check if "Bob" is an element of \(S\).
Given:
\(S = \{\text{Alice}, \text{Bob}, \text{Charles}\}\), element \(b = \text{Bob}\)
Since Bob is listed in \(S\),$$
b \in S
$$
Answer: Bob is an element of set \(S\)
Example 2:
Given set \(R = \{101, 102, 103, 104\}\), determine if 105 is an element of \(R\).
Given: \(R = \{101, 102, 103, 104\}\), element \(x = 105\)
Since 105 is not in \(R\),$$
105
otin R
$$
Answer: 105 is not an element of \(R\)
Example 3:
A bank has a staff set \(F = \{201, 202, 203, 204, 205\}\). If employee 203 transfers out, is 203 still an element of \(F\)?
Given: \(F = \{201, 202, 203, 204, 205\}\), employee \(e = 203\) leaves
After removal:$$
F' = \{201, 202, 204, 205\}
$$$$
203
otin F'
$$
Answer: 203 is no longer an element of the updated set \(F'\)
Example 4:
Set \(G = \{\text{HR Manager}, \text{Accountant}, \text{Clerk}\}\). Is "Secretary" an element of \(G\)?
Given: \(G\) as above, element \(s = \text{Secretary}\)
Since Secretary is not listed,$$
s
otin G
$$
Answer: Secretary is not an element of \(G\)
Example 5:
If set \(L\) represents employees eligible for leave, and employee 110 is in \(L\), write the membership statement.
Given: \(L\) and element \(110\)$$ 110 \in L $$ Answer: Employee 110 is an element of set \(L\)
An empty set or null set is a set that contains no elements. It is denoted by \(\emptyset\) or \(\{\}\). In HR, an empty set might represent a scenario where no employees meet a specific criterion, such as no staff eligible for a particular training or benefit.
The empty set is always a subset of any set.
Example 1:
A university HR department defines set \(X\) as employees who joined after the year 2025. Since it is currently 2024, find \(X\).
Given: No employees joined after 2025$$ X = \emptyset $$ Answer: \(X\) is the empty set
Example 2:
Set \(Y\) contains employees who have been with a company for over 50 years. If no employee qualifies, express \(Y\).
Given: No employee with >50 years service$$ Y = \emptyset $$ Answer: \(Y\) is the empty set
Example 3:
In a SACCO, set \(Z\) is employees who have defaulted on loans but also have perfect attendance. If no employee meets both criteria, find \(Z\).
Given: No employees satisfy both conditions$$ Z = \emptyset $$ Answer: \(Z\) is the empty set
Example 4:
If set \(A\) represents employees who earn less than Ksh 10,000 per month in a bank where minimum salary is Ksh 12,000, find \(A\).
Given: Minimum salary is Ksh 12,000$$ A = \emptyset $$ Answer: \(A\) is the empty set
Example 5:
Set \(B\) includes employees who have both managerial and internship status simultaneously. If no employee holds both, express \(B\).
Given: No overlap between managers and interns$$ B = \emptyset $$ Answer: \(B\) is the empty set
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Create a free accountThis chapter introduced the fundamental concepts of set theory beginning with basic definitions such as a set, an element, and the empty set. It then explored the primary operations on sets including union, which combines all elements from two sets, intersection, which identifies common elements, difference, which finds elements in one set but not the other, and symmetric difference, which includes elements in either set but not in both. The use of Venn diagrams was explained as a visual tool to represent these set operations clearly. Finally, the chapter demonstrated how set theory can be applied in various practical business and statistical contexts to organize data and solve problems efficiently. Together, these topics provide a comprehensive foundation for understanding and applying set theory in business mathematics and statistics.
In a human resource database, set \(A\) represents employees in the Marketing department with 15 members, and set \(B\) represents employees in the Finance department with 20 members. If no employee works in both departments, find the number of employees in the union \(A \cup B\). (2 marks)
In a company, set \(A\) contains 30 employees trained in software skills, and set \(B\) contains 25 employees trained in communication skills. If 10 employees are trained in both skills, find the total number of employees in either training program. (3 marks)
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