Intro to Business Math, Logic, Sets, and Basic Algebra

Scope and Importance of Business Mathematics

Business mathematics is not just about numbers; it is the application of mathematical tools to business problems. It bridges the gap between theoretical mathematics and practical management. In the context of TU BBA, this unit lays the groundwork for all subsequent quantitative analysis.

Why is it important?

1. Decision Making: Managers use math to choose between alternatives (e.g., which product line to expand).
2. Forecasting: Predicting future sales or costs using historical data.
3. Optimization: Maximizing profit or minimizing cost with limited resources.
4. Communication: Providing a precise, universal language for financial reporting.

mindmap
  root((Business Mathematics))
    Decision Making
      Cost-Benefit Analysis
      Risk Assessment
    Forecasting
      Trend Analysis
      Regression
    Optimization
      Linear Programming
      Marginal Analysis
    Communication
      Financial Statements
      Data Visualization

Mathematical Logic

Logic is the study of valid reasoning. In business, we use logic to evaluate arguments, such as "If demand increases, then price should increase."

Propositions and Connectives

A proposition is a statement that is either true (T) or false (F).

• Negation ($\sim p$): The opposite of $p$.
• Conjunction ($p\wedge q$): True only if both $p$ and $q$ are true.
• Disjunction ($p\vee q$): True if at least one is true.
• Implication ($p⟹q$): False only if $p$ is true and $q$ is false.

Truth Tables

We use truth tables to determine the truth value of compound statements.

Worked Example: Let $p$: "The company has sufficient capital." (True) Let $q$: "The project is profitable." (False)

Evaluate: $p\wedge \sim q$

1. $p=T$
2. $q=F$
3. $\sim q=T$ (Negation of False is True)
4. $p\wedge \sim q=T\wedge T=T$

Interpretation: The company has capital, and the project is not profitable. This statement is logically true based on the premises.

Set Theory

Sets are collections of distinct objects. In business, sets represent groups of customers, products, or employees.

Basic Terminology

• Element: A member of a set.
• Subset ($A\subseteq B$): Every element of $A$ is in $B$.
• Universal Set ($U$): The set of all elements under consideration.
• Empty Set ($\varnothing$): A set with no elements.

Set Operations

1. Union ($A\cup B$): Elements in $A$ or $B$ or both.
2. Intersection ($A\cap B$): Elements in both $A$ and $B$.
3. Complement ($A^{\prime }$): Elements in $U$ but not in $A$.
4. Difference ($A-B$): Elements in $A$ but not in $B$.

Worked Example: Customer Segmentation

Assume a bank has 100 customers.

• Set $A$: Customers who use Credit Cards. $|A|=60$.
• Set $B$: Customers who use Online Banking. $|B|=40$.
• Set $A\cap B$: Customers who use both. $|A\cap B|=20$.

Question: How many customers use at least one of the services?

Solution: Using the Principle of Inclusion-Exclusion: $$|A\cup B|=|A|+|B|-|A\cap B|$$ $$|A\cup B|=60+40-20=80$$

So, 80 customers use at least one service. The number of customers using neither is: $$|U|-|A\cup B|=100-80=20$$

Algebraic Structures

Algebraic structures define the rules for operations on sets. The most common in business math are Groups and Fields.

Groups

A set $G$ with an operation $\cdot$ is a group if:

1. Closure: $a\cdot b\in G$.
2. Associativity: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3. Identity: There exists $e$ such that $a\cdot e=a$.
4. Inverse: For every $a$, there is $a^{-1}$ such that $a\cdot a^{-1}=e$.

Example: The set of integers $\mathbb{Z}$ under addition is a group.

• Identity: 0
• Inverse of $x$: $-x$

Fields

A field is a set with two operations (addition and multiplication) that satisfy group properties for both, plus distributivity.

• Real numbers $\mathbb{R}$ form a field.
• This allows us to divide (except by zero) and solve linear equations, which is crucial for financial modeling.

Functions and Relations

A function $f:A\to B$ assigns exactly one element of $B$ to each element of $A$.

• Domain: The set of inputs ($A$).
• Codomain: The set of possible outputs ($B$).
• Range: The actual set of outputs.

Types of Functions

1. One-to-One (Injective): Different inputs give different outputs.
2. Onto (Surjective): Every element in the codomain is mapped to.
3. Bijective: Both one-to-one and onto.

Business Application: A pricing function $P(Q)$ maps Quantity ($Q$) to Price ($P$). If the function is one-to-one, each price corresponds to a unique quantity, making demand analysis straightforward.

In the real world

1. eSewa and Khalti (Set Theory): Digital wallets use set operations to manage user permissions. For example, the set of "Verified Users" ($V$) and the set of "High-Value Users" ($H$). The intersection $V\cap H$ identifies users eligible for premium cashback offers. If a user is in $V$ but not $H$, they get standard rates. This logic is hardcoded into the backend to automate marketing decisions.

2. Daraz (Logical Implications): Daraz’s recommendation engine uses logical rules. Rule: IF (User viewed "Laptop") AND (User added "Mouse" to cart) THEN (Recommend "Laptop Stand"). This is a conjunction ($p\wedge q$) leading to an action. If the logic is flawed (e.g., using OR instead of AND), the system might recommend a mouse stand to someone who only looked at a laptop, reducing conversion rates.

3. Ncell and NTC (Algebraic Structures): Billing systems rely on the field properties of real numbers. Calculating call charges involves multiplication (rate $\times$ time) and addition (summing multiple calls). The ability to divide total cost by number of minutes to find an "average cost" relies on the multiplicative inverse property of the real number field. Without these algebraic guarantees, billing discrepancies would be impossible to resolve mathematically.

Exam tip

For TU BBA exams, Unit 1 questions are usually direct.

1. Truth Tables: You will likely be asked to construct a truth table for a given logical expression. Practice identifying the main connective first.
2. Set Problems: Venn diagram problems are standard. Always draw the diagram first. Label the intersection clearly. Use the formula $|A\cup B|=|A|+|B|-|A\cap B|$ for two sets. For three sets, use the inclusion-exclusion principle: $$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$$
3. Definitions: Be precise. Do not just say "a group is a set." Say "a set with an operation satisfying closure, associativity, identity, and inverse properties."
4. Functions: Be ready to determine if a given relation is a function by checking if any input has multiple outputs.