Course Mathematics-L1

Course Summary 

Part 1: Real Numbers & Number Sets

  • Real Numbers: Used to measure continuous quantities such as distance, duration, or temperature.

  • Classification of Subsets:

    • Natural Numbers: Positive integers used for counting.

    • Integers: Whole numbers including positive integers, negative integers, and zero.

    • Decimal Numbers: Numbers with a finite number of digits after the decimal point.

    • Rational Numbers: Numbers that can be expressed as a fraction of two integers.

    • Irrational Numbers: Numbers that cannot be written as a quotient of two integers (e.g., square root of two, Pi, Euler's number).

  • Properties & Bounded Sets:

    • Absolute Value: Measures the distance of a number from zero, always yielding a non-negative value.

    • Floor Function: The greatest integer less than or equal to a given real number.

    • Maximum / Minimum: Respectively the greatest and least elements belonging to a set.

    • Upper / Lower Bound: An upper bound is greater than or equal to all set elements; a lower bound is less than or equal to all set elements.

    • Supremum / Infimum: The supremum is the least upper bound; the infimum is the greatest lower bound.

Part 2: Mathematical Logic

  • Statements:

    • Proposition: A statement that is strictly either true or false.

    • Axiom: A fundamental proposition accepted as true without proof.

    • Theorem: A statement that has been proven true.

  • Logical Connectives:

    • Negation: Gives the opposite truth value of a proposition.

    • Conjonction ("AND"): True only when both linked propositions are true.

    • Disjunction ("OR"): True if at least one of the linked propositions is true.

    • Implication ("IF... THEN"): Connects a premise to its logical consequence.

    • Equivalence ("IF AND ONLY IF"): True when both propositions share the same truth value.

  • Quantifiers: Express whether a property holds for all elements (universal quantifier) or at least one element (existential quantifier).

Part 3: Vectors & Vector Spaces

  • Geometric Vectors: Represented by an arrow defined by magnitude (length), direction (supporting line), and sense (arrowhead orientation).

  • Vector Space Structure: An abstract set equipped with two fundamental operations:

    1. Vector Addition: Associative, commutative, containing an additive identity (zero vector), and an inverse for every element.

    2. Scalar Multiplication: Multiplication of a vector by a real number, satisfying distributive and scaling properties.

  • Key Concepts:

    • Linear Combination: A vector produced by multiplying vectors by scalars and adding them together.

    • Linear Independence (Free Sets): No vector in the collection can be written as a combination of the others.

    • Linear Dependence (Bound Sets): At least one vector can be written as a combination of the others.

    • Basis: A minimal linearly independent set that spans the entire vector space.

    • Dimension: The unique number of vectors forming any basis of the vector space.