Open and closed doors on the straight

Include and exclude endpoints. How do interval and inequality symbols change?

Explore the controls and explain what changed in your own words.

The intervals will be represented as a subset of the real numbers or better known as the straight line. These subsets are infinite, because no matter how many smaller intervals you take, they will still be infinite. Unlike the set of natural numbers or integers.

Example 1.

Limitations

Definition (Interval): an interval is a subset of the straight line (real numbers).

Definition (Inferiorly limited interval): an interval is inferiorly limited, if we have any real number $x$ such that it is greater than or equal to another real number $a$.

$$\forall x \in \mathbb{R}, ~x\geq a$$
Example 1.

Definition (Upper limited interval): an interval is superior limited, if we have any real number $x$ such that it is less than or equal to another real number $b$.

$$\forall x \in \mathbb{R}, ~x\leq b$$
Example 1.

Definition (Limited Interval): An interval is limited if it is limited inferiorly and limited superiorly.

$$\forall x \in \mathbb{R}, ~a \leq x\leq b$$
Example 1.

Definition (Unlimited interval): an interval is unlimited if it is neither limited inferiorly nor limited superiorly.

$$\forall x \in \mathbb{R}$$
Example 1.

Limited open ranges

Here we will discuss the ranges of real numbers. They are fundamental for inequalities, among other issues. Below we present the definition of an open interval.

Definition: Let $a,b$ be real numbers, that is, $a,b\in \mathbb{R}$. We say that a range is open when we have $(a,b) \subset \mathbb{R}$.

When the range is open it means that the numbers $a,b$ do not belong to the range.

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

Notation: $(~)$ parentheses.

Example 1.

Example for open ranges

For $a=1,$ $b=5,$ determine the open range. Then the range is given by:

$$(1,5)\in \mathbb{R}$$

Examples of some numbers belonging to these ranges.

  • $1\notin (1,5)$:
    • $1$ does not belong to the range $(1,5)$
  • $5\notin (1,5)$:
    • $5$ does not belong to the range $(1,5)$
  • $2\in (1,5)$:
    • $2$ belongs to the range $(1,5)$
  • $\dfrac{3}{2}\in (1,5)$:
    • $\dfrac{3}{2}$ belongs to the range $(1,5)$
  • $\pi\in (1,5)$:
    • $\pi$ belongs to the range $(1,5)$
  • $e \in (1,5)$:
    • $e$ belongs to the range $(1,5)$

Limited closed ranges

Below we present the definition of a closed interval.

Definition: Let $a,b$ be real numbers, that is, $a,b\in \mathbb{R}$. We say that an interval is closed when we have $[a,b] \subset \mathbb{R}$.

When the range is closed it means that the numbers $a,b$ do not belong to the range.

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

Notation: $[~]$ square brackets.

Example 1.

Example for closed intervals

For $a=1,$ $b=5,$ determine the closed range. Then the range is given by:

$$[1,5]\in \mathbb{R}$$

Examples of some numbers belonging to these ranges.

  • $1\in [1,5]$:
    • $1$ does not belong to the range $[1,5]$
  • $5\in [1,5]$:
    • $5$ does not belong to the range $[1,5]$
  • $2\in [1,5]$:
    • $2$ belongs to the range $[1,5]$
  • $\pi\in [1,5]$:
    • $\pi$ belongs to the range $[1,5]$
  • $e \in [1,5]$:
    • $e$ belongs to the range $[1,5]$

Limited semi-open ranges

Below we present the definition of a semi-open interval.

Definition: Let $a,b$ be real numbers, that is, $a,b\in \mathbb{R}$. We say that an interval is semi-open when we have $[a,b] \subset \mathbb{R}$.

When the range is semi-open it means that the numbers $a,b$ do not belong to the range.

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

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

Notation: $[~)$ or $(~]$ parentheses and brackets.

Example 1.

Example for semi-open intervals

For $a=1,$ $b=5,$ determine the half open interval. Then the range is given by:

$$(1,5]\in \mathbb{R}$$

Examples of some numbers belonging to these ranges.

  • $1\notin (1,5]$:
    • $1$ does not belong to the range $(1,5]$
  • $5\in (1,5]$:
    • $5$ does not belong to the range $(1,5]$
  • $2\in (1,5]$:
    • $2$ belongs to the range $(1,5]$
  • $\pi\in (1,5]$:
    • $\pi$ belongs to the range $(1,5]$
  • $e \in (1,5]$:
    • $e$ belongs to the range $(1,5]$