5. Intervalos

1. Intervalos- nuevo concepto

Para nombrar a muchos números tenemos los elementos matemáticos que te vamos a presentar:

LOS INTERVALOS

Imagen de elaboración propia con Canva. *Intervalos numéricos* (CC BY-NC-SA)

Son elementos que engloban a una cantidad infinita de números reales (recuerda, tanto racionales como irracionales).

Los intervalos deben tener dos extremos (uno inicial y otro final), aunque haya otros que tienen el $ \pm \infty \ $. Estas se llamarían SEMIRRECTAS.

Según esté incluido o no el extremo, hay varios tipos de intervalos:

ABIERTO, (a, b)

Serán todos los números reales comprendidos entre a y b sin incluirlos.

Por ejemplo:

El intervalo (3,6) es el conjunto de números de la recta real entre el 3 y el 6 sin estar incluidos ni el 3 ni el 6.

Se simboliza con círculos abiertos

Intervalo abierto ej1 — Imagen de elaboración propia (CC BY-NC-SA)

CERRADO, [a, b]

Serán todos los números reales comprendidos entre a y b ambos incluidos.

Por ejemplo:

El intervalo [1,4] es el conjunto de números de la recta real entre el 1 y el 4 incluyendo el 1 y el 4.

Se simboliza con círculos rellenos

Intervalo cerrado ej 2 — Imagen de elaboración propia (CC BY-NC-SA)

SEMIABIERTO o SEMICERRADO, [a, b) o (a,b]

Serán todos los números reales comprendidos entre a y b incluyendo al número con el corchete. [, ]

Por ejemplo:

El intervalo (-1,7] es el conjunto de números de la recta real entre el -1 y el 7. El -1 no está incluido pero el 7 sí.

Int semiabierto ej 3 — Imagen de elaboración propia (CC BY-NC-SA)

El intervalo [2,5) es el conjunto de números de la recta real entre el 2 y el 6. El 5 no está incluido pero el 2 sí.

Inter semiabierto ej 4 — Imagen de elaboración propia (CC BY-NC-SA)

SEMIRRECTAS

Serán aquellas donde uno de los extremos es $ + \infty \ $ o $ - \infty \ $

En estos casos, el $ \infty \ $ nunca está incluido

La semirrecta [1, $ + \infty \ $) es el conjunto de números de la recta real mayores o igual que 1

Se representa donde esté el $ \infty \ $ con una flecha.

interv semirrecta ej 5 — Imagen de elaboración propia (CC BY-SA)

Aquí puedes ver un vídeo para comprenderlo mejor:

Vídeo de Tuto mate. Intervalos. Tipos. Representación (CC BY)

2. Formas de representar un intervalo

Con una representación gráfica

Se representa con un segmento destacando sus extremos. Si es abierto, se representa con círculos abiertos y si es cerrado, con círculos rellenos.

Si se trata de una semirrecta, se indicará con una flecha en el lado del -$ \infty \ $ o +$ \infty \ $

Esto lo hemos visto en el apartado anterior.

Como conjuntos de números reales

Se escribe entre llaves { , } con desigualdades. Es un modo más "matemático"

{$ { x \in \mathbb{R} / a\leq x \leq b} $ } $\Longleftrightarrow $ $[a, b] $

{$ { x \in \mathbb{R} / a < x \leq b} $ } $\Longleftrightarrow $ $(a, b] $

{$ { x \in \mathbb{R} / a\leq x < b} $ } $\Longleftrightarrow $ $[a, b) $

{$ { x \in \mathbb{R} / a < x < b} $ } $\Longleftrightarrow $ $(a, b) $

{$ { x \in \mathbb{R} / a < x} $ } $\Longleftrightarrow $ {$ { x \in \mathbb{R} / x > a} $ } $\Longleftrightarrow $ $(a, +\infty ) $

{$ { x \in \mathbb{R} / a \leq x} $ } $\Longleftrightarrow $ {$ { x \in \mathbb{R} / x \geq a} $ } $\Longleftrightarrow $ $[a, +\infty ) $

{$ { x \in \mathbb{R} / a > x} $ } $\Longleftrightarrow $ {$ { x \in \mathbb{R} / x < a} $ } $\Longleftrightarrow $ $(-\infty, a ) $

{$ { x \in \mathbb{R} / a \geq x} $ } $\Longleftrightarrow $ {$ { x \in \mathbb{R} / x \leq a} $ } $\Longleftrightarrow $ $(-\infty, a] $

Con intervalos

Se expresa con los paréntesis ( ) cuando el extremo no esté incluido. Se dice abierto.

Se expresa con los corchetes [ ] cuando el extremo esté incluido. Se dice cerrado.

$ ( a, b) $ - Intervalo abierto entre los números $a$ y $b$ sin estar ningún extremo incluido.

$ [a, b] $ - Intervalo cerrado entre los números $a$ y $b$ con los dos extremos incluidos.

$ [a, b) $ - Intervalo semiabierto (o semicerrado) entre los números $a$ y $b$ incluyendo solo a $a$ y no a $b$.

$ (a, b] $ - Intervalo semiabierto (o semicerrado) entre los números $a$ y $b$ incluyendo solo a $b$ y no a $a$.

3. Comprueba lo visto con GeoGebra

Mueve los puntos del intervalo y comprueba lo visto anteriormente.

https://www.geogebra.org/m/bw85KpRz (Ventana nueva)

Antonio%20Omatos%20Soria,https%3A//ggbm.at/161828,Representaci%F3n%20de%20intervalos%20en%20la%20recta%20real,1,Autor%EDa

Actividad%20no%20completada,Actividad%20superada.%20Puntuaci%F3n%3A%20%25s,Actividad%20no%20superada.%20Puntuaci%F3n%3A%20%25s,Guardar%20la%20puntuaci%F3n

4. Practica

Escribe de las otras dos maneras la expresión de cada intervalo:

Modo abreviado

a) [ 2, 5]

b) (-1, 4]

c) [0, 3)

d) (- $ \infty\ $, 3]

e) (-1, $ \infty\ $)

Modo gráfico

a)	Imagen de elaboración propia (CC BY-NC-SA)
b)	Imagen de elaboración propia (CC BY-NC-SA)
c)	Imagen de elaboración propia (CC BY-NC-SA)
d)	Imagen de elaboración propia (CC BY-NC-SA)
e)	Imagen de elaboración propia (CC BY-NC-SA)

Modo conjunto-desigualdades

a)	{$ { x \in \mathbb{R} / 2\leq x \leq 5} $ } Audio de elaboración propia (CC BY-NC-SA)
b)	{$ { x \in \mathbb{R} / -1 < x \leq 4} $ } Audio de elaboración propia (CC BY-NC-SA)
c)	{$ { x \in \mathbb{R} / 0\leq x < 3} $ } Audio de elaboración propia (CC BY-NC-SA)
d)	{$ { x \in \mathbb{R} / x \leq 3} $ } $\Longleftrightarrow $ {$ { x \in \mathbb{R} / 3 \geq x} $ } Audio de elaboración propia (CC BY-NC-SA)
e)	{$ { x \in \mathbb{R} / -1 < x} $ } $\Longleftrightarrow $ {$ { x \in \mathbb{R} / x > -1} $ } Audio de elaboración propia (CC BY-NC-SA)

5. Practica

Selecciona la imagen correcta para cada intervalo.

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