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Technology, Environment and Society Module - Coggle Diagram
Technology, Environment and Society Module
Unit I. Number Sets
Natural, Integer, Rational, Irrational, and Real Numbers
o Natural numbers are non-negative integers (0,1,2,3,…).
o Rational numbers can be expressed as fractions (ab\frac{a}{b}ba), while irrational numbers cannot be written as such (e.g., π\piπ, 2\sqrt{2}2).
o Real numbers include both rational and irrational numbers, forming the complete set of numbers we use in daily life.
Operations and Properties
o Basic properties like commutativity, associativity, and distributivity govern operations like addition, subtraction, multiplication, and division.
o Specific rules exist for each number set: natural numbers do not allow division by zero, while integers allow operations with negative numbers.
o The properties of operations apply differently in each set, e.g., multiplication of fractions follows specific rules.
Inequalities, Real Intervals, and Absolute Value
o Inequalities show relationships between numbers (e.g., a>ba > ba>b means aaa is greater than bbb).
o Real intervals are sets of numbers on the real line, expressed as open or closed (e.g., [a,b][a, b][a,b] or (a,b)(a, b)(a,b)).
o The absolute value of a number is its distance from zero on the real line, regardless of its sign (e.g., ∣−5∣=5|-5| = 5∣−5∣=5).
Properties of Exponents and Radicals
o Exponent properties include rules like am⋅an=am+na^m \cdot a^n = a^{m+n}am⋅an=am+n and (am)n=am⋅n(a^m)^n = a^{m \cdot n}(am)n=am⋅n.
o Radicals represent roots (e.g., a\sqrt{a}a), and have properties similar to fractional exponents.
o Radicals can be rationalized (eliminate the radical in the denominator) using conjugates or multiplying by an appropriate form.
Rationalization
o Rationalization involves eliminating roots from the denominator of a fraction by multiplying by an expression that simplifies the root.
o This process is commonly used with fractions containing square or cube roots in the denominator.
o Rationalization is useful for simplifying calculations and obtaining more manageable expressions.
Applications: Modular Arithmetic
o Modular arithmetic studies operations on integers according to a modulus nnn, with the notation amod na \mod namodn.
o It is useful for solving divisibility problems, like finding the remainder of a division.
o It is applied in cryptography, number theory, and computer algorithms.
Unit II. Algebraic Expressions
Algebraic Expressions
o These are combinations of variables, constants, and algebraic operations.
o They can be simplified by combining like terms and applying operations.
o Factorization and expansion are important techniques when working with algebraic expressions.
Polynomials and Operations with Polynomials
o A polynomial is a sum of monomials and is classified by the number of terms (monomial, binomial, trinomial).
o Operations include addition, subtraction, multiplication, and division of polynomials.
o The Remainder Theorem states that when dividing a polynomial by x−ax - ax−a, the remainder is equal to the value of the polynomial evaluated at x=ax = ax=a.
Special Products
o These are products that can be expanded easily without needing to perform all the multiplication.
o Examples: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2 and a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)a2−b2=(a+b)(a−b).
o They facilitate expression simplification and are fundamental in polynomial factorization.
Rational Algebraic Expressions
o These are fractions where both the numerator and denominator are polynomials.
o They must be simplified by removing common factors between the numerator and denominator.
o Operations of addition, subtraction, multiplication, and division can be performed on algebraic fractions.
Quotient of Polynomials
o The quotient of polynomials can be obtained by dividing polynomials using methods like synthetic or long division.
o The Remainder Theorem and the Factor Theorem are useful in this process.
o The result of the division may be a quotient plus a remainder.
Unit III. Matrices and Determinants
Definition
o A matrix is a rectangular array of numbers arranged in rows and columns.
o It is classified by its size, represented as m×nm \times nm×n (m rows and n columns).
o Matrices can represent systems of equations, geometric transformations, and other problems.
Order of a Matrix
o The order of a matrix refers to the number of rows and columns it has: m×nm \times nm×n.
o Square matrices have the same number of rows and columns (m=nm = nm=n).
o Rectangular matrices have a different number of rows and columns.
Order of a Matrix
o Special matrices include the identity matrix (with 1's on the diagonal and 0's elsewhere) and the diagonal matrix (with non-zero values only on the diagonal).
o The null matrix has all its elements equal to zero.
o The transpose of a matrix is obtained by swapping its rows and columns.
Special Matrices
o Special matrices include the identity matrix (with 1's on the diagonal and 0's elsewhere) and the diagonal matrix (with non-zero values only on the diagonal).
o The null matrix has all its elements equal to zero.
o The transpose of a matrix is obtained by swapping its rows and columns.
Algebraic Sum
o Matrices of the same order are added by adding their corresponding elements.
o Matrix addition is commutative and associative.
o The zero matrix acts as the neutral element in addition.
Inverse Matrix
o The inverse matrix of a square matrix AAA is the matrix A−1A^{-1}A−1 such that A⋅A−1=IA \cdot A^{-1} = IA⋅A−1=I, where III is the identity matrix.
o Not all matrices have an inverse, only non-singular matrices (with a non-zero determinant).
o The inverse matrix can be found using methods such as Cramer's rule or the adjoint formula.
Elementary Operations with Rows and Columns
o Elementary operations include multiplying a row or column by a scalar, exchanging rows or columns, and adding one row or column to another.
o These operations are used to solve systems of equations using Gaussian elimination.
o They are essential for finding the inverse of matrices or calculating determinants.
Unit IV. Vectors
Geometric Vectors
o Vectors have magnitude (length) and direction.
o They can be added and subtracted graphically using the parallelogram method or the head-to-tail method.
o Vectors can be unit vectors (magnitude 1) or directed in any way in space.
Components in R2R^2R2 and R3R^3R3
o A vector in R2R^2R2 has two components (x, y), and in R3R^3R3 it has three (x, y, z).
o The magnitude of a vector is calculated using the Pythagorean theorem.
o The representation of vectors in coordinates makes algebraic and graphical operations easier.
Introduction: Trigonometry-Basic Concepts. Graphs
o Trigonometry is used to describe angles and relationships on the Cartesian plane.
o Vectors are graphically represented as arrows in space.
o The coordinates of a vector in R2R^2R2 or R3R^3R3 correspond to its components.
Graphical Representation
o Vectors are represented as arrows with a specific direction on the Cartesian plane.
o Graphical operations include vector addition, magnitude calculation, and the angle between vectors.
o Geometric transformations like translations and rotations can be described using vectors.
Analytical and Graphical Operations
o Operations include addition, subtraction, scalar multiplication, and dot product of vectors.
o The analytical representation uses components to describe operations.
o Graphically, transformations and results can be visualized.
Norm of a Vector
o The norm of a vector is its length or magnitude, calculated as ∥v∥=x2+y2|v| = \sqrt{x^2 + y^2}∥v∥=x2+y2 in R2R^2R2 and ∥v∥=x2+y2+z2|v| = \sqrt{x^2 + y^2 + z^2}∥v∥=x2+y2+z2 in R3R^3R3.
o The norm is used to measure the "intensity" or "size" of the vector.
o A unit vector has a norm equal to 1.
Dot Product
o The dot product between two vectors is defined as v⋅w=∥v∥∥w∥cos(θ)v \cdot w = |v| |w| \cos(\theta)v⋅w=∥v∥∥w∥cos(θ), where θ\thetaθ is the angle between them.
o It is a measure of how parallel two vectors are.
o It is used in vector projection and calculating the angle between vectors.
Unit V. Propositional Logic
Proposition
o A proposition is a statement that can be either true or false.
o Propositions are represented by letters such as PPP, QQQ.
o The truth value of a proposition is essential for logical reasoning.
Logical Connectives
o Logical connectives like "and" (∧\land∧), "or" (∨\lor∨), "not" (¬\neg¬), and "if...then" (→\rightarrow→) are used to combine propositions.
o They allow the construction of compound propositions from simple ones.
o They are fundamental in mathematical logic and computational theory.
Compound Propositions
o A compound proposition is a combination of simple propositions connected by logical connectives.
o Examples: P∧QP \land QP∧Q (P and Q), P→QP \rightarrow QP→Q (if P then Q).
o The evaluation of its truth value depends on the truth values of the propositions it is composed of.
Logical Laws
o Logical laws are rules that describe how logical connectives behave (e.g., De Morgan’s law, distributive law).
o They help simplify and manipulate logical expressions.
o They are essential for solving problems in mathematical logic.
Valid Deductive Reasoning
o Deductive reasoning is reasoning where, if the premises are true, the conclusion must also be true.
o Examples of valid inferences include modus ponens and modus tollens.
o They are used to prove theorems in mathematics and logic.
Unit VI. Sets
Element and Membership
o A set is a collection of distinct objects, called elements.
o The symbol ∈\in∈ denotes membership, meaning an element belongs to a set (e.g., a∈Aa \in Aa∈A means "a is in set A").
o Sets can be finite (with a limited number of elements) or infinite (with no limit on elements).
Definition by Comprehension and Extension
o A set can be defined by extension (listing all its elements explicitly, e.g., A={1,2,3}A = {1, 2, 3}A={1,2,3}).
o It can also be defined by comprehension (describing a property that all elements satisfy, e.g., A={x∣x is an even number}A = { x \mid x \text{ is an even number} }A={x∣x is an even number}).
o The comprehension method is useful for infinite or complex sets.
Inclusion
o If all elements of set AAA are also in set BBB, then AAA is a subset of BBB, written as A⊆BA \subseteq BA⊆B.
o If A⊆BA \subseteq BA⊆B but A≠BA \neq BA=B, then AAA is a proper subset of BBB.
o The universal set contains all elements under consideration, and every set is a subset of the universal set.
Union
o The union of two sets AAA and BBB, written A∪BA \cup BA∪B, includes all elements that are in either AAA, BBB, or both.
o The union operation is commutative (A∪B=B∪AA \cup B = B \cup AA∪B=B∪A) and associative ((A∪B)∪C=A∪(B∪C)(A \cup B) \cup C = A \cup (B \cup C)(A∪B)∪C=A∪(B∪C)).
o It is used in probability, database queries, and set operations in mathematics.
Intersection
o The intersection of two sets AAA and BBB, written A∩BA \cap BA∩B, includes all elements that are common to both sets.
o If A∩B=∅A \cap B = \emptysetA∩B=∅ (empty set), then AAA and BBB are disjoint (have no common elements).
o The intersection operation is also commutative and associative.