Interval Notation

Interval notation is a method for representing continuous sets of real numbers by defining their boundaries. Although having the appearance of arranged pairs, it describes every number between those endpoints rather than identifying specific values. They describe the range of values that fall within the boundaries, providing a succinct way to express inequalities or systems of inequalities.

By grouping a variety of numbers into a single representation, interval notation facilitates the understanding of the numerical representation in mathematical terms. For example, suppose we want to represent the set of real numbers {x | -2 < x < 5} using an interval. This can be expressed using the interval notation (-2, 5).

In interval notation, the set of real numbers can be represented as (-∞, ∞)

The numbers that make up the set can be used to categorize intervals. While some sets may or may not contain the endpoints listed in the notation, others may. Generally speaking, there are three different kinds of intervals:

• Closed Interval
   
• Open Interval
   
• Half-Open Interval

Closed Interval 

This type of interval includes the endpoints of the inequality. For example, the set {x | -3 ≤ x ≤ 1} contains the endpoints -3 and 1. For this, the closed interval notation is [-3,1].

Open Interval

This type of interval excludes the endpoints of the inequality. For example, the set {x | -3 < x < 1} does not contain the endpoints -3 and 1. This is expressed using open interval notation: (-3, 1).

Half-Open Interval

This type of interval contains only one endpoint of the inequality. For instance, the set {x | -3 ≤ x < 1} contains the endpoint -3. This is expressed using half-open interval notation: [-3, 1).

Interval notation for real numbers indicates a continuous range of values by stating the lower and upper bounds of the range, as well as whether each end is included or excluded. An excluded endpoint indicates that the boundary value itself is not a part of the set, whereas an included endpoint indicates that it is.

For example, every real number x that satisfies 2 ≤ x < 5 is described by the interval from 2 to 5 with 2 included but 5 excluded. Similarly, x < 0 describes the set of all negative numbers less than zero (with zero excluded and negative infinity understood as the lower bound). Lastly, by permitting x to vary from negative infinity to positive infinity—excluding both infinities since they are unreachable—the complete real line is captured.

By identifying the endpoints of an inequality and indicating whether they are included or excluded, interval notation provides a simplified method of describing all of its solutions. We specify an open endpoint to indicate exclusion (< or >) and a closed endpoint to indicate inclusion (⩽ or ⩾) instead of using inequality signs.

For instance, "minus 3 less than or equal to x less than 4" becomes "from negative three to four, including -3 but excluding 4," and "x greater than 2" becomes "from 2 to infinity, excluding 2." Since infinity and negative infinity are unreachable, they are always regarded as open endpoints.

Interval notation is used to represent a set of real numbers within a range. Interval notation uses parentheses or brackets to list all elements between two endpoints. The set-builder notation, for instance, {x ∈ R ∣ 2 < x < 5} symbolizes all real numbers between 2 and 5, excluding those that are not included. The endpoint is included when it is enclosed in square brackets [], while parentheses () indicate exclusion. Interval notation is used in algebra and calculus to represent domains, ranges, and inequalities' solutions. It is particularly helpful for representing continuous sets.

By listing the endpoints of each interval and specifying whether each end is included, interval notation for a function's domain names the continuous range of permitted x-values. Any unbounded direction is described by negative or positive infinity and is always excluded; an endpoint is included if the function exists exactly at that boundary and excluded if it does not.

As an example of the function f(x)=1x - 3, is defined for all real numbers except for 𝑥=3, meaning that its domain spans from negative infinity to three, but not beyond, and then from slightly above three to positive infinity. This succinct, accurate explanation is very helpful when drawing or analyzing graphs, as well as in algebra and calculus.

All the potential output values, or y-values, that a function can generate can be succinctly described using interval notation for range. Interval notation depicts the range as a continuous set, as opposed to listing individual values or employing inequalities. Endpoints are included using brackets [] and excluded using parentheses ().

For instance, if the output of a function can begin at a specific value and increase to infinity, we use a parenthesis for infinity and a bracket for the starting value because infinity cannot be reached. For example, since squaring any real number yields a result of zero or greater, the range of the function 𝑓(𝑥) = x2 is [0, ∞). Another example is 𝑓(𝑥) = 1x, whose range is expressed as (−∞, 0) ∪ (0, ∞) since its output is never zero. In higher-level mathematics, such as algebra and calculus, this range-writing technique is particularly helpful in clearly defining the behavior of functions.

To represent the interval notation for various interval types, we can use specific guidelines and symbols. Let's examine the various symbols that are available for writing a specific kind of interval.
We employ the following notations for various intervals:

• [ ]: This square bracket is used when both endpoints are included in the set.

• ( ): When both endpoints are not included in the set, this round bracket is utilized.

• ( ]: This semi-open bracket is used when the left endpoint is not in the set and the right endpoint is.

• [ ]: This semi-open bracket is also used when the set's left endpoint is included and its right endpoint is excluded.

Using particular visual cues to indicate whether the endpoints are included or excluded is necessary when representing various interval types on number lines. This makes it easier to understand the kind of interval being discussed. 

Open Interval 

Draw a line between two points and use open (hollow) circles at the endpoints, a and b, to show open intervals on a number line. This indicates that the interval does not include either endpoint. 
For instance, (–3, 2) → hollow circles at –3 and 2, joined by a solid line.

Closed Interval [a, b]

A solid line between two points indicates a closed interval; filled (solid) dots at a and b indicate that both endpoints are included.
For instance, [–1, 4] → solid dots at –1 and 4 joined by a solid line.

Half-Open Interval [a, b) or (a, b]

In a half-open interval, one endpoint is included and the other is excluded. For [1, 7], the line is shaded between a filled circle at 1 (included) and an open circle at 7 (excluded). For (–5, 2]: a filled circle at 2 (included) and an open circle at –5 (excluded), with shading once more between. Differentiating which endpoints are a part of the interval is made easier by these visual cues.

An inequality can be converted to interval notation by following a straightforward, step-by-step procedure. Find the inequality symbol first. You will use parentheses in the interval if the inequality uses < or > because this indicates that the endpoint is not included. The endpoint is included if the inequality uses ≤ or ≥, so square brackets will be used. Next, ascertain the interval's direction. The interval extends infinitely in one direction and includes infinity if the inequality only involves one comparison, such as 𝑥 > 2. Since infinity is not a precise number, it is always expressed in parentheses. 

For instance, 𝑥 > 2 turns into (2, ∞). The smaller number is placed first in a bounded interval formed by a compound inequality, such as −3 < 𝑥 ≤ 4, with the brackets or parentheses adjusted according to whether the endpoints are included. Firstly, sort out the number values from smallest to largest and write it under the proper notation. One can easily recognize the numerical range.

In everyday situations, such as weather forecasts or safety instructions, interval notation is a useful method of representing value ranges that makes numerical limits understandable and straightforward.

• Ranges of Temperature
  
  According to a weather report, the temperature during the day will be between 18 and 25 degrees Celsius. That range is represented in interval notation as [18, 25], which indicates that every temperature x that satisfies 18 ≤ x ≤ 25 °C is anticipated. You can decide whether to bring a light jacket or a T-shirt by quickly looking at it.

• Road Speed Limits
  
  Highway speed limits can be represented using interval notation. The legal range is [60, 100] if a highway strictly caps traffic at 100 km/h while enforcing speeds of at least 60 km/h. You would write (60, 100), omitting 60, if the lower limit were advisory rather than mandatory, such as when speeds over 60 km/h are advised but not required. The absolute maximum and the safe minimum are both made evident by interval notation.

• Appropriate Blood Pressure Ranges
  
  According to doctors, normal blood pressure ranges from 90 to 120 mmHg. This is [90, 120] as an interval, so any value between 90 ≤ x ≤ 120 is considered healthy. Patients can easily determine if their numbers are within or outside this target range.

• Age Restrictions on Services
  
  A student event is open to those aged 13 to 17 (but not 18), according to [13, 17]. In other words, people who are 13, 14, 15, 16, and 17 years old are permitted, but people who are younger or older are not. This keeps it clear who is eligible to take part.

• Safe pH Ranges in Swimming Pools
  
  Pool managers keep the pH between 7.2 and 7.8 to safeguard swimmers and equipment. Every water sample reading x with 7.2 ≤ x ≤ 7.8 is acceptable, as indicated by the notation [7.2, 7.8]. Pool chemistry can be checked and adjusted quickly and methodically by using this interval.