MTH 101 CHAPTER 1-6

FEDERAL UNIVERSITY, DUTSIN-MA

Overview

• Faculty: Physical Sciences

• Department: Mathematics

• Course Code: MTH 101

• Course Title: Elementary Mathematics I (Algebra and Trigonometry)

• Units: 2

Course Description

• This course aims to provide a solid foundation in algebra and trigonometry, covering fundamental topics, including:

  1. Elementary Set Theory: Concepts of subsets, union, intersection, complements, and Venn diagrams.

  2. Number Systems and Mathematical Induction: Exploration of natural numbers, integers, rational, and irrational numbers; principles of mathematical induction.

  3. Real Sequences and Series: Study of sequences (Arithmetic and Geometric Progressions).

  4. Theory of Quadratic Equations: Properties and solutions, including the quadratic formula.

  5. Binomial Theorem: Formulae for expanding powers of binomials.

  6. Complex Numbers: Properties and operations; De Moivre's theorem, and nth roots of unity.

  7. Circular Measure: Trigonometric functions of angles in radians.

  8. Trigonometric Functions: Properties, addition and factor formulas.

Chapter One: Elementary Set Theory

1.1 Sets

• Definition: A set is a collection of well-defined and distinct objects.

• Notation: Capital letters denote sets, small letters denote elements.

• Example: {1,2,3} is the same as {3,1,2}.

1.2 Membership and Representation

• Membership: Denoted by 𝑎 ∈ 𝐴 (𝑎 is an element of set 𝐴).

• Representation: Two forms:

  • Tabular Form: List all members. Example: P = {3,5,7,11,13,17,19}.

  • Set Builder Notation: {𝑥 | 𝑥 satisfies condition}.

1.3 Subset

• Definition: A set 𝐵 is a subset of 𝐴 if all members of 𝐵 are also members of 𝐴 (𝐵 ⊆ 𝐴).

• Proper Subset: Denoted 𝐵 ⊂ 𝐴 if 𝐵 is strictly contained within 𝐴.

• Example: {3,5,7} ⊂ {3,5,7,9}.

1.4 Equality of Sets

• Condition: Two sets are equal if every element of 𝐴 is in 𝐵 and vice versa.

• Example: Compare sets for equality:

  • 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑}, 𝐵 = {𝑑, 𝑎, 𝑐, 𝑒} → 𝐴 ≠ 𝐵

  • 𝐹 = {1,2,5}, 𝐻 = {5,1,2} → 𝐹 = 𝐻.

1.5 Universal, Unit, and Empty Sets

• Universal Set (𝒰): The set containing all objects under consideration.

• Unit Set: Contains a single element. Example: {a}.

• Empty Set (∅): Contains no elements. Example: {𝑥: 𝑥 is even prime > 2}.

1.6 Union, Intersection, and Complement of Sets

• Union (𝐴 ∪ 𝐵): All elements in A or B.

• Intersection (𝐴 ∩ 𝐵): Elements common to A and B.

• Complement (𝐴ᶜ): Elements not in A.

• Examples:

  • If 𝐴 = {2,3,4} and 𝐵 = {3,4,6,7}, then 𝐴 ∪ 𝐵 = {2,3,4,6,7}.

1.7 Difference and Disjoint Sets

• Difference (𝐵 - 𝐴): Elements in B but not in A.

• Disjoint Sets: If 𝐴 ∩ 𝐵 = ∅, then A and B are disjoint.

1.8 Cardinality of a Set

• Definition: The number of elements in a set, denoted n(𝑆).

• Example: n(𝐴) = 26 for the letters of the English alphabet.

1.9 Venn Diagrams

• Usage: Representing sets and their relationships.

Exercises

1. Prove that the following statements are equivalent:

   • (i) 𝐴 ⊆ 𝐵

   • (ii) 𝐴 ∩ 𝐵 = 𝐴

   • (iii) 𝐴 ∪ 𝐵 = 𝐵

2. Show that the sets 𝐴\𝐵 and 𝐵\𝐴 are disjoint.

3. Use Venn diagrams to illustrate various set operations.

Chapter Two: Number Systems and Mathematical Induction

2.1 Number Systems

• Natural Numbers (ℕ): Counting numbers.

• Integers (ℤ): Includes negative numbers.

• Rational Numbers (ℚ): Expressible as a fraction p/q.

• Irrational Numbers (ℝ \ ℚ): Cannot be expressed as fractions.

• Complex Numbers (ℂ): a + bi, where i = √−1.

2.2 Mathematical Induction

• Definition: Method for proving statements for all natural numbers.

• Principle: Step one: Show true for n=1; step two: Show true for n=k implies true for n=k+1.

• Examples: Sum of first n even numbers, sum of first n natural numbers.

Exercises

Prove simple propositions using induction.

Note: Continue with similar structured notes for additional chapters, maintaining clarity, definitions, examples, and exercises for effective studying.