BED 122 — Business Mathematics: Lecture Note

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LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Course code: BED 122 Course title: Business Mathematics Course credit load: (2 Units C: LH 30) List of Series SERIES 1 Mathematics and Symbolic Logic SERIES 2 Matrices, Determinants, and Vectors SERIES 3 Complex Numbers and Coordinate Geometry SERIES 4 Sequences, Series, and Calculus SERIES 5 Optimization and Linear Programming 1 Mathematics and Symbolic Logic Series Outline • Part 1: Foundations of Symbolic Logic o Sub-Part 1.1: Basic concepts of logic o Sub-Part 1.2: Logical operators and truth tables o Sub-Part 1.3: Applications of symbolic logic in reasoning • Part 2: Deductive and Inductive Reasoning o Sub-Part 2.1: Principles of deductive reasoning o Sub-Part 2.2: Principles of inductive reasoning o Sub-Part 2.3: Problem-solving approaches using reasoning • Part 3: Sets and Their Applications o Sub-Part 3.1: Definition and types of sets o Sub-Part 3.2: Set operations and Venn diagrams o Sub-Part 3.3: Applications of sets in mathematics and business This outline gives us three main Parts, which conforms to the guideline for a 2-credit course. Introduction

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Background Mathematics is not only about numbers, it is also about reasoning and structure. Symbolic logic provides a systematic way of thinking, allowing us to analyse arguments and solve problems with clarity. In business mathematics, logical reasoning and the study of sets form the foundation for more advanced topics such as matrices, calculus, and optimisation. This Series will help you build these essential skills. Series Aim The aim of this Series is to equip you with the ability to apply symbolic logic and reasoning techniques, and to use sets effectively in mathematical and business contexts. Series Synopsis We shall commence this Series by exploring the foundations of symbolic logic, where you will learn about logical concepts, operators, and truth tables. Next, we will move on to deductive and inductive reasoning, examining how these approaches support problem-solving in mathematics and business. Finally, we will conclude with sets and their applications, focusing on definitions, operations, and practical uses in everyday problem-solving. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1 define the basic concepts of symbolic logic, • SLO 1.2 explain the use of logical operators and construct truth tables, • SLO 1.3 apply deductive reasoning to solve structured problems, • SLO 1.4 analyse situations using inductive reasoning for generalisation, and • SLO 1.5 demonstrate how to perform set operations and interpret Venn diagrams in mathematical and business contexts. 1.1 Foundations of Symbolic Logic 1.1.1 Basic concepts of logic Logic is the study of reasoning. In mathematics, symbolic logic refers to the use of symbols to represent logical expressions and arguments. This allows us to analyse statements systematically and determine whether they are valid or invalid. A proposition is a declarative statement that can be either true or false, but not both. For example, “The sky is blue” is a proposition because it can be evaluated as true or false. Fill in the blank: A statement that can be either true or false, but not both, is called a ________.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Diagram showing examples of propositions and non-propositions Caption: Figure 1.1 Examples of propositions and non-propositions 1.1.2 Logical operators and truth tables In symbolic logic, logical operators are symbols used to connect propositions and form compound statements. The most common operators are: • Negation (¬): reverses the truth value of a proposition. • Conjunction (∧): represents “and”, true only if both propositions are true. • Disjunction (∨): represents “or”, true if at least one proposition is true. • Implication (→): represents “if…then”, true except when the first proposition is true and the second is false. • Biconditional (↔): represents “if and only if”, true when both propositions have the same truth value. A truth table is a tabular representation that shows all possible truth values of compound statements. It is a powerful tool for analysing logical expressions. Fill in the blank: The operator that represents “if…then” is called ________. Caption: Table 1.1 Truth table for conjunction (p ∧ q)

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS p q p∧q T T T T F F F T F F F F 1.1.3 Applications of symbolic logic in reasoning Symbolic logic is widely applied in mathematics, computer science, and business decision-making. For example, in programming, logical operators are used to control the flow of decisions. In business, logical reasoning helps managers evaluate scenarios and make consistent decisions. Consider this example: A company decides to invest in a project if the market demand is high and production costs are low. Symbolic logic allows us to represent this decision as: • p: Market demand is high • q: Production costs are low • Decision rule: p ∧ q Fill in the blank: The decision rule “p ∧ q” will only be true if ________.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Caption: Box 1.1 Application of symbolic logic in business decision-making Pilot questions 1.1 1. Which of the following is a proposition? a. Close the door b. The sun rises in the east c. What time is it d. Please help me 2. Which logical operator represents “and”? a. ∨ b. ∧ c. → d. ¬ 3. In a truth table, the conjunction p ∧ q is true when: a. Both p and q are false b. At least one of p or q is true c. Both p and q are true d. p is true and q is false 4. Which operator reverses the truth value of a proposition? a. ∧ b. ¬

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS c. → d. ↔ 5. A business decision rule expressed as p ∧ q will be true only when: a. p is true and q is false b. p is false and q is true c. both p and q are true d. both p and q are false 1.2 Deductive and Inductive Reasoning 1.2.1 Principles of deductive reasoning Deductive reasoning is a logical process where conclusions are drawn from general statements or premises. If the premises are true, the conclusion must also be true. For example, if “All managers are employees” and “James is a manager”, then we can deduce that “James is an employee”. Deductive reasoning is often used in mathematics and business to ensure accuracy and reliability. Fill in the blank: If all managers are employees and James is a manager, then James is an ________. Diagram showing deductive reasoning with premises and conclusion Caption: Figure 1.2 Deductive reasoning structure 1.2.2 Principles of inductive reasoning Inductive reasoning involves drawing general conclusions from specific observations. Unlike deduction, inductive reasoning does not guarantee certainty, but it provides probable conclusions.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS For example, if a business notices that sales increase every December, it may induce that December is a peak season. Fill in the blank: Inductive reasoning moves from ________ observations to general conclusions. Image showing inductive reasoning with examples from business trends Caption: Plate 1.1 Inductive reasoning in business 1.2.3 Problem-solving approaches using reasoning Both deductive and inductive reasoning are essential in problem-solving. Deduction ensures logical certainty, while induction helps in predicting patterns and making informed decisions. In business mathematics, combining both approaches allows managers to evaluate risks and opportunities effectively. Fill in the blank: Deductive reasoning ensures ________, while inductive reasoning provides probable conclusions. Caption: Box 1.2 Comparing deductive and inductive reasoning

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