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Mathematics · 100 Level
This is the complete lecture note for MTH 101 Elementary Mathematics I, prepared for Mathematics students at 100 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Series 1: Foundations of Set Theory and Venn Diagrams Series outline 1.1 Basic Concepts of Sets 1.1.1 Definition of a set 1.1.2 Methods of describing sets 1.1.3 Types of sets (finite, infinite, empty, universal) 1.2 Subsets and Set Operations 1.2.1 Subsets and proper subsets 1.2.2 Union of sets 1.2.3 Intersection of sets 1.2.4 Complements of sets 1.3 Venn Diagrams and Applications 1.3.1 Representation of sets using Venn diagrams 1.3.2 Two-set and three-set Venn diagrams 1.3.3 Problem-solving with Venn diagrams 1.4 Practical Applications of Set Theory 1.4.1 Everyday examples of sets and subsets 1.4.2 Use of Venn diagrams in probability and logic Introduction In everyday life, we often group objects, ideas, or people into categories, whether it is arranging books on a shelf, sorting clothes by colour, or classifying students by their courses. These groupings are examples of what mathematicians call sets, and they form the foundation of many areas of mathematics. Understanding sets and how they interact is essential because they provide the language and structure for more advanced topics such as algebra, probability, and logic. This Series will
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS therefore introduce you to the basic principles of set theory and the use of Venn diagrams, which are powerful tools for visualizing relationships between sets. The aim of this Series is to equip you with the ability to define sets clearly, perform operations on them, and represent these operations using Venn diagrams. By the end of the Series, you should be able to apply these concepts to solve mathematical problems and recognize their relevance in real-world contexts. We shall commence this Series by exploring the basic concepts of sets, including their definitions, methods of description, and types. Next, we will examine subsets and set operations such as union, intersection, and complements. Following that, we will move on to Venn diagrams, where you will learn how to represent and interpret sets visually, including two-set and three-set diagrams. Finally, we will wrap up by considering practical applications of set theory, showing how these ideas can be used in everyday situations and in fields such as probability and logic. Series Learning Outcomes Upon completing this Series, you should be able to: 2.1 define the concept of a set and distinguish between different types of sets, 2.2 explain the idea of subsets and demonstrate how to perform basic set operations such as union, intersection, and complements, 2.3 illustrate relationships between sets using Venn diagrams and interpret two-set and three-set representations, 2.4 apply set theory and Venn diagrams to solve practical problems in everyday contexts and in mathematics. 1.1 Basic Concepts of Sets 1.1.1 Definition of a set A set is a well-defined collection of distinct objects, called elements, which are grouped together as a single entity. The term “well-defined” means that there should be no ambiguity about whether an object belongs to the set or not. For example, the set of vowels in the English alphabet can be written as {a, e, i, o, u}. Sets are usually denoted by capital letters such as A, B, or C, while their elements are listed inside curly brackets. If an element belongs to a set, we say it is a member of that set.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: A ______ is a well-defined collection of distinct objects regarded as a single entity. Figure 1.1 diagram of set notation vowels 1.1.2 Methods of describing sets There are two common methods of describing sets: Roster method: This involves listing all the elements of the set explicitly inside curly brackets. For example, the set of even numbers less than 10 can be written as {2, 4, 6, 8}. Set-builder notation: This involves describing the property that characterises the elements of the set. For example, the same set of even numbers less than 10 can be written as {x | x is an even number less than 10}. Fill in the blank: The ______ method lists all elements explicitly, while the ______ notation describes them by a property. Box 1.1 Methods of describing sets Content: Roster method: Listing elements explicitly. Set-builder notation:
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Table 1.1: set theory roster vs set-builder notation Feature Roster (Tabular) Method Set-Builder Notation Definition Lists every single element within braces. Describes the properties elements must satisfy. Best Used For Small, finite sets. Large, infinite, or complex sets. Visual Style {1, 2, 3, 4, 5} {x | x is a positive integer < 6} 1.1.3 Types of sets Sets can be classified into different types based on their properties: Finite set: A set with a countable number of elements. Example: {1, 2, 3, 4}. Infinite set: A set with an uncountable or unlimited number of elements. Example: {1, 2, 3, …}. Empty set (null set): A set with no elements, denoted by {} or Ø. Universal set: A set that contains all possible elements under consideration in a particular context. Fill in the blank: A set with no elements is called the ______ set. Table 1.2 Types of sets with examples Type of set Definition Example Finite set Countable number of elements {1, 2, 3, 4} Infinite set Uncountable or unlimited number of elements {1, 2, 3, …} Empty set No elements {} or Ø Universal set All elements under consideration U = {all letters of alphabet}
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Pilot questions 1.1 Which of the following best defines a set? a) A random group of objects b) A well-defined collection of distinct objects c) A list of repeated items d) A group with no rules Which method of describing sets involves listing all elements explicitly? a) Set-builder notation b) Roster method c) Universal method d) Infinite method What symbol is commonly used to denote the empty set? a) {} or Ø b) U c) ∩ d) ∪ Which of the following is an example of an infinite set? a) {1, 2, 3, 4} b) {a, e, i, o, u} c) {1, 2, 3, …} d) {} The set of all letters in the English alphabet is an example of which type of set? a) Finite set b) Infinite set c) Universal set d) Empty set 1.2 Subsets and Set Operations 1.2.1 Subsets and proper subsets A subset is a set in which all its elements are contained within another set. If every element of set A is also an element of set B, then A is said to be a subset of B, written as A ⊆ B. A proper subset is a subset that is strictly smaller than the set it belongs to, meaning A ⊂ B if A is contained in B but A ≠ B.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: If every element of set A is also in set B, then A is a ______ of Figure 1.2 Venn diagram showing set A inside set B to illustrate subset 1.2.2 Union of sets The union of two sets is the set containing all elements that belong to either of the sets. It is denoted by A ∪ B. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}. Fill in the blank: The union of sets A and B is written as ______.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Figure 1.3 Union of two sets Venn diagram” 1.2.3 Intersection of sets The intersection of two sets is the set containing all elements that are common to both sets. It is denoted by A ∩ B. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}. Fill in the blank: The intersection of sets A and B is written as ______.
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