The document defines several mathematical concepts:
1. Sets are collections of elements with similar characteristics that are considered as objects. Elements can include numbers, colors, letters, etc.
2. Set operations like union, intersection, and difference allow combining sets to obtain new sets.
3. Inequalities are relations of order between two values, indicating whether one value is less than, greater than, or equal to another.
4. The absolute value of a real number represents the distance between that number and zero on the number line.
Unidad 2: Números Reales y Plano Numérico
Maickel Pineda
CI: 30.304.460
Aula 0103
Universidad Politécnica Territorial Del estado Lara
"Andrés Eloy Blanco"
Programa Nacional De Formación en Agroalimentación
Este archivo te servirá para recordar y manejar mejor temas sobre los números reales y conjuntos ademas de valor absoluto, así como también una serie de ejercicio resueltos que te ayudaran a entender mejor la teoría
Unidad 2: Números Reales y Plano Numérico
Maickel Pineda
CI: 30.304.460
Aula 0103
Universidad Politécnica Territorial Del estado Lara
"Andrés Eloy Blanco"
Programa Nacional De Formación en Agroalimentación
Este archivo te servirá para recordar y manejar mejor temas sobre los números reales y conjuntos ademas de valor absoluto, así como también una serie de ejercicio resueltos que te ayudaran a entender mejor la teoría
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Unit 8 - Information and Communication Technology (Paper I).pdf
Matematica
1. República Bolivariana de Venezuela
Universidad Politécnica territorial Andrés Eloy blanco
Estado LARA
Integrate
WILKELyS Suarez
C.i: 29654718
SECCION: 0102
Matemática
2. Definición De Los Conjunto
Un conjunto es una colección de elementos con características similares considerada en sí
misma como un objeto. Los elementos de un conjunto, pueden ser las
siguientes: personas, números, colores, letras, figuras, etc. Se dice que
un elemento (o miembro) pertenece al conjunto si está definido como incluido de algún
modo dentro de él.
Ejemplo: el conjunto de los colores del arcoíris es:
AI = {rojo, naranja, amarillo, verde, azul, añil, violeta}
Un conjunto suele definirse mediante una propiedad que todos sus elementos poseen.
Por ejemplo, para los números naturales, si se considera la propiedad de ser un número
primo, el conjunto de los números primos es:
P = {2, 3, 5, 7, 11, 13,…}
3. Operaciones con Conjunto
Las operaciones con conjuntos también conocidas como álgebra de conjuntos, nos permiten
realizar operaciones sobre los conjuntos para obtener otro conjunto. De las operaciones con
conjuntos veremos los siguientes unión, intersección, diferencia, diferencia simétrica y
complemento.
Ejemplo
Dados dos conjuntos A={1,2,3,4,5} y B={4,5,6,7,8,9} la unión de estos conjuntos será
A∪B={1,2,3,4,5,6,7,8,9}. Usando diagramas de Ven se tendría lo siguiente:
4. Números Reales
El conjunto de los números reales (denotado por R) incluye tanto a los números racionales,
(positivos, negativos y el cero) como a los números irracionales;1
y en otro
enfoque, trascendentes y algebraicos. Los irracionales y los trascendentes2
(1970) no se pueden
expresar mediante una fracción de dos enteros con denominador no nulo; tienen infinitas cifras
decimales aperiódicas, tales como √5, π, o el número real log2, cuya trascendencia fue enunciada
por Euler en el siglo XVIII.2
5. Desigualdad
Es una relación de orden que se da entre dos valores cuando estos son distintos (en caso
de ser iguales, lo que se tiene es una igualdad).
La notación a < b significa a es menor que b;
La notación a > b significa a es mayor que b
Estas relaciones se conocen como desigualdades estrictas, puesto que a no puede ser
igual a b; también puede leerse como "estrictamente menor que" o "estrictamente mayor
que".
La notación a ≤ b significa a es menor o igual que b;
La notación a ≥ b significa a es mayor o igual que b;
La notación a ≠ b significa que ha no es igual a b. Tal expresión no indica si uno es
mayor que el otro, o siquiera si son comparables.
Generalmente se tienden a confundir los operadores según la posición de los elementos
que se están comparando; didácticamente se enseña que la abertura está del lado del
elemento mayor. Otra forma de recordar el significado, es recordando que el signo
señala/apunta al elemento menor.
6. Definición de valor absoluto
Para cualquier número real X, el valor absoluto de X denota por X se define como:3
IXI ={
𝑋, 𝑠𝑖 𝑋 ≥ 0
𝑋, 𝑠𝑖 𝑋 < 0
El valor absoluto de X es siempre un número positivo o cero pero nunca negativo:
cuando X es un número negativo (X < 0) entonces su valor absoluto es necesariamente
positivo .( IXI = -X > 0 ).
Desde un punto de vista geométrico, el valor absoluto de un número real puede verse como
la distancia que existe entre ese número y el cero. De manera general, el valor absoluto entre
la diferencia de dos números es la distancia entre ellos.
7. Desigualdades con valor absoluto
Para cualquier valor positivo de a:
Es equivalente a (esta regla también aplica a )
Es equivalente a x −a o x a (esta regla también aplica a )
X puede ser una variable o una expresión algebraica.
Ejemplo:
Apliquemos lo que ya sabes sobre resolver ecuaciones que contienen valores absolutos y lo que sabes
sobre desigualdades para resolver desigualdades que contienen valores absolutos. Empecemos con
una desigualdad simple.
La desigualdad dice, “el valor absoluto de x es menor o igual a 4.” Si se te pide resolver x, quieres
encontrar los valores de x que están a 4 unidades o menos de 0 en la recta numérica. Podrías empezar
imaginando la recta numérica y los valores de x que satisfacen esta ecuación.
8. 4 y −4 están a cuatro unidades del 0, entonces son soluciones. 3 y −3 también son soluciones porque
cada uno de estos valores está a menos de cuatro unidades del 0. Al igual que el 1 y el −1, el 0.5 y
el −0.5, etc. — hay un número infinito de valores de x que satisfacen la desigualdad.
La gráfica de esta desigualdad tendrá dos círculos cerrados, en 4 y en −4. La distancia entre estos dos
círculos en la recta numérica está coloreada de azul porque estos son los valores que satisfacen la
ecuación.
La solución se puede escribir de esta manera: −4 x 4.