Concepto y Tipos de Intervalos

Understanding Intervals in Mathematics

Definition of Interval

• An interval is defined as the set of real numbers between two endpoints, referred to as the extremes of the interval. For example, the numbers between 2 and 10 are included in this definition.

Types of Intervals

Open Intervals

• An open interval does not include its endpoints but includes all values in between. This is denoted using circular parentheses. For instance, numbers greater than 5 and less than 10 can be expressed as (5, 10).

• The representation on a number line involves placing open circles at the endpoints (5 and 10) to indicate that these values are not included in the interval.

Closed Intervals

• A closed interval includes both endpoints along with all values in between, represented by square brackets. For example, numbers greater than or equal to 5 and less than or equal to 10 can be written as [5, 10].

• On a number line, closed intervals are depicted with solid circles at both ends (indicating inclusion of endpoints). This visually represents that both extremes are part of the set.

Semi-open Intervals

• A semi-open interval includes one endpoint while excluding the other. It uses a combination of circular and rectangular parentheses; for example, 3, 9) indicates that 3 is included while 9 is not. [

• When graphed on a number line, only one endpoint will have a solid circle (for included) while the other will have an open circle (for excluded). This distinction helps clarify which values belong to the set.

Infinite Intervals

• An infinite interval extends indefinitely in one direction; it may include positive or negative infinity as an endpoint but cannot be fully quantified numerically. The specifics were not detailed within this segment but suggest further exploration into their properties would follow later in the video.

Understanding Open and Closed Intervals in Mathematics

Introduction to Intervals

• The concept of intervals can be infinite at one or both ends, leading to open or closed extremes. This means that intervals can be semi-open or fully open.

Example of a Closed Interval

• An example is the set of numbers less than or equal to 9. The interval notation for this would be from negative infinity (open) to 9 (closed), represented as (-∞, 9].

Visual Representation on a Number Line

• On a number line, the right endpoint (9) is marked with a solid line indicating it is included, while the left endpoint (negative infinity) has no line since it is not defined.

Another Example: Open Interval

• For numbers greater than 2, starting from 3 and extending to positive infinity, the interval notation would be (2, +∞). Here, 2 is not included in the interval.

Visual Representation of Open Interval

• On the number line for this example, an open circle indicates that 2 is not included in the interval. A right-pointing arrow signifies that values extend towards positive infinity.