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 \(\{ x \mid -2 < x < 5 \} \) 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 \(\{ x \mid -3 \le x \le 1 \} \) 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 \(\{ x \mid -3 < x < 1 \} \) 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 \(\{ x \mid -3 \le x < 1 \} \) 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 with \(2 \le x < 5\) lies in the interval \([2, 5)\), including 2 but excluding 5. Similarly, \(x < 0\) represents all negative numbers. Allowing x to vary from \(-\infty {\text { to } }+\infty\) (excluding infinities) describes the entire real line.

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 example, \(-3 \le x < 4\) means the interval from -3 to 4, including -3 but excluding 4, and \(x > 2 \) means from 2 to infinity, excluding 2. Infinity and negative infinity are always 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.

Set-builder notation, like  \(\{ x \in \mathbb{R} \mid 2 < x < 5 \} \), represents all real numbers between 2 and 5. Square brackets [] include an endpoint, while parentheses () exclude it. Interval notation is widely used in algebra and calculus to show domains, ranges, and solutions, especially for 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.

For example, the function \(f(x) = \frac{1}{x-3}\)is defined for all real numbers except \(x = 3\). Its domain is \((-\infty, 3) \cup (3, \infty)\). This clear explanation helps when analyzing graphs or solving problems 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 example, when a function’s output starts at a number and goes up to infinity, we use a bracket [ ] for the starting number and a parenthesis ( ) for infinity because infinity can’t be reached.

• For \(f(x) = x^2\), squaring any number gives zero or more, so its range is \([0, \infty)\).
   
• For \(f(x) = \frac{1}{x}\), the output is never zero, so the range is \((-\infty, 0) \cup (0, \infty)\).

Using brackets and parentheses this way helps students easily understand how functions behave in algebra and calculus.
 

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, the inequality \(x > 2\) converts to the interval \((2, \infty)\). For a compound inequality like \(-3 < x \le 4\), first identify the smallest and largest values, then use parentheses or brackets depending on whether the endpoints are excluded or included. This process converts inequalities into interval notation, making it easy to see the numerical range.

Interval notation is used to represent a range of numbers clearly and concisely. Mastering it helps students to understand the relationship between inequalities and number ranges, use proper symbols like brackets and parentheses, and express mathematical intervals accurately. Here are a few tips and tricks to master interval notation. 

• Understand the use of brackets and parentheses, the square bracket [] is used to represent endpoints that are included, while parentheses ( ) mean they are not included. 
   
• Always connect interval notation to its inequality form to its inequality form for better understanding. For example, \({3 < x ≤ 7}\) is written as \((3, 7]\). This helps students to identify which endpoints are included or excluded. 
   
• Intervals are always written in ascending order from the smaller number to the larger number. For example, \([2, 10], (5, 15]\). 
   
• When solving the word problems, always highlight the keywords like including, up to, or less than. For including, use square brackets [], and for less than or up to use parentheses ( ). 
   
• Use number lines to visualize which points are included or excluded. Closed circles represent brackets and open circles represent parentheses. 

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.

• In weather forecast, interval notation helps show the expected temperature range of the day. For example, if the temperature is between \(18^\circ \text C\) and \(25^\circ \text C\), it is written as \([18, 25]\), which means the temperature x such that \(18 ≤ x ≤ 25\). 
   
• Highway speed limits can be represented using interval notation. For example, A highway may require speed from \(60\ \text{km/h} \) to \(100\ \text{km/h} \), then it can be expressed as \([60, 100]\). The absolute maximum and the safe minimum are both made evident by interval notation.
   
• The interval notation is used to show normal medical ranges. For example, the normal blood pressure is \([90, 120]\), so any value between 90 ≤ x ≤ 120 is considered healthy.
   
• Events or services have certain age limit. For example, a youth workshop for ages 13-17 is written as \([13, 17]\). 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.
   
• Banks have interset rates for saving account or loans as a range. For example, a saving account might offer \([3 \%, 5\%]\) annual interest, letting customers know the minimum and maximum possible returns. 
   
• Interval notation is used to indicate safe dosage range for medicine. For example, if the recommended dose of a drug is between 50 mg and 100 mg, it can be represented as \([50, 100] \text{ mg } \).