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MTH 103

Basic Mathematics

TU BITM · Semester 1 · BITM curriculum and programme regulation

Requirement
required
Credits
3
Study Guide
9 units available

Chapter notes

Authoritative study guide across 9 syllabus units

Open a unit to read its study notes.

Unit 1: Set Theory and Real Number System· 6 LHs· ~8 marks

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Part 1: Core Concepts (2 Marks)

  • Power Set P(A)\mathcal{P}(A): The set of all subsets of a given set AA. If ∣A∣=n|A| = n, then the cardinality of its power set is strictly ∣P(A)∣=2n|\mathcal{P}(A)| = 2^n.
  • De Morgan’s Laws: Fundamental duality theorems in set algebra stating (A∪B)′=A′∩B′(A \cup B)' = A' \cap B' and (A∩B)′=A′∪B′(A \cap B)' = A' \cup B'.
  • Dedekind Completeness of Real Numbers: The foundational property asserting that every non-empty set of real numbers bounded from above possesses a least upper bound (supremum) in R\mathbb{R}.
  • Absolute Value Inequality: For any x,a∈Rx, a \in \mathbb{R} with a>0a > 0, ∣x∣≤a  ⟺  −a≤x≤a|x| \le a \iff -a \le x \le a, and ∣x∣≥a  ⟺  x≤−a or x≥a|x| \ge a \iff x \le -a \text{ or } x \ge a.

Part 2: Structural Frameworks (10 Marks)

Structural Hierarchy of the Real and Complex Number Systems

N⊂W⊂Z⊂Q⊂R⊂C\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}
Number Classification Standard Symbol Mathematical Formalization Defining Canonical Examples
Natural Numbers N\mathbb{N} {1,2,3,4,… }\{1, 2, 3, 4, \dots\} 1,9,1081, 9, 108
Whole Numbers W\mathbb{W} {0}∪N={0,1,2,3,… }\{0\} \cup \mathbb{N} = \{0, 1, 2, 3, \dots\} 0,7,240, 7, 24
Integers Z\mathbb{Z} {…,−3,−2,−1,0,1,2,3,… }\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\} −45,0,12-45, 0, 12
Rational Numbers Q\mathbb{Q} $\left{\frac{p}{q} ;\middle ; p, q \in \mathbb{Z},; q \neq 0,; \gcd(p, q) = 1\right}$
Irrational Numbers Q′\mathbb{Q}' Non-terminating, non-repeating decimal numbers 2,π,e,ϕ\sqrt{2}, \pi, e, \phi
Real Numbers R\mathbb{R} Q∪Q′\mathbb{Q} \cup \mathbb{Q}' (All points along the continuous real line) −3,0,3.14159-\sqrt{3}, 0, 3.14159
Imaginary Numbers I\mathbb{I} {bi∣b∈R,b≠0,i=−1}\{bi \mid b \in \mathbb{R}, b \neq 0, i = \sqrt{-1}\} 5i,−2.7i5i, -2.7i
Complex Numbers C\mathbb{C} {z=x+iy∣x,y∈R,i2=−1}\{z = x + iy \mid x, y \in \mathbb{R}, i^2 = -1\} 3+4i,−1+i33 + 4i, -1 + i\sqrt{3}

Part 3: Distinction Tables

Basis of Comparison Rational Numbers (Q\mathbb{Q}) Irrational Numbers (Q′\mathbb{Q}')
Fractional Form Can be expressed as pq\frac{p}{q} where p,q∈Z,q≠0p, q \in \mathbb{Z}, q \ne 0 Cannot be expressed as an integer ratio
Decimal Expansion Terminating or non-terminating repeating (periodic) Non-terminating and non-repeating (aperiodic)
Algebraic Character Closed under addition, subtraction, multiplication, division Not closed under basic arithmetic operations
Examples 34,0.333…,−5,0\frac{3}{4}, 0.333\dots, -5, 0 2,3,π,e\sqrt{2}, \sqrt{3}, \pi, e

Part 4: Worked Problems & Exam Traps

Worked Set Problem (Three Intersecting Sets)

Problem: In a cohort of 100 IT students: 50 know C, 40 know Java, 35 know Python. 20 know C and Java, 15 know Java and Python, 18 know C and Python, and 10 know all three. Find the number of students who know none of these languages.

∣C∪J∪P∣=∣C∣+∣J∣+∣P∣−∣C∩J∣−∣J∩P∣−∣C∩P∣+∣C∩J∩P∣=50+40+35−20−15−18+10=125−53+10=82\begin{aligned} |C \cup J \cup P| &= |C| + |J| + |P| - |C \cap J| - |J \cap P| - |C \cap P| + |C \cap J \cap P| \\ &= 50 + 40 + 35 - 20 - 15 - 18 + 10 \\ &= 125 - 53 + 10 = 82 \end{aligned}
Students knowing none=∣U∣−∣C∪J∪P∣=100−82=18\text{Students knowing none} = |U| - |C \cup J \cup P| = 100 - 82 = 18

TU Exam Trap: Remember the Principle of Inclusion-Exclusion alternates signs! The three-set intersection ∣A∩B∩C∣|A \cap B \cap C| must be added, not subtracted, at the final step.