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# Bounds|Definition & Meaning

## Definition

A** bound in mathematics** is a **limit** on the size or value of a quantity. Bounds are often used to describe the **range of possible values** for a variable, or the **range of possible outcomes** for a function. Bounds can be used to find the **maximum or minimum value** of a function, or to prove that a function is continuous.

A **mathematical bound** is a limit that a function or variable can **approach** but never beyond. An example is shown below:

**Figure 1:** Bounds using **inequalities**

## Applications of Bounds

The concept of a **bound** is crucial in many areas of mathematics, such as **analysis** and **optimization.** Analysis typically uses bounds to show the **existence** of **certain boundaries. **To find the best solution to a problem, optimization specialists employ **bounds** as a **tool.**

## Types of Bounds

There are two types of bounds that are commonly used in mathematics, namely u**pper bounds** and **lower bounds.** The **upper bound** of a function or variable is its **highest value,** whereas the **lower bound** of a variable is its **minimum value. **Both of these types are further explained in next paragraphs.

### (a) Upper Bound

An **upper Bound** is a type of bound defined by a **number** that is **greater than** **or equal** to all the possible values of a function, variable or set. **Upper bounds** are widely used in mathematical **optimization** scenarios when the goal is to identify the **largest value** of a function.

**Upper bounds** can be formulated using a variety of methods, including **derivative-based methods (e.g. gradient descent)** and **convex optimization. **In some cases, it is possible to prove that a **certain** **value** is an **upper** **bound****.** This can be done using mathematical proof techniques, such as **induction****. **

It is often useful to have **multiple** **upper** **bounds****,** so that the best possible solution can be found. In many cases, the best upper bound is the one that is **tight****est,** **meaning** that it is closest to the **actual** **value****.**

There are two** types** of upper bounds: (a) **tight** upper bounds and (b) **loose** upper bounds. A **tight** **upper** **bound** is the best possible upper bound on a quantity, while a **loose** **upper** **bound** is not necessarily the best possible upper bound on a quantity.

### (b) Lower Bound

A **lower** **bound** is a number that is numerically **lower** **than** **or** **equal** **to** to all the possible values of a function, variable or set. **For** **example****,** if we have a number x, then a lower bound of x would be any number that is less than or equal to x. There are many ways to **formulate** **lower** **bounds****.** One way is to use a **graph****.** Another way is to use **inequalities****.**

Inequalities are **mathematical** **statements** that say that one number is greater than or equal to another number. Continuing with the **same** **example****,** let‘s say we have a **number**** x**, and we want to find a lower bound for it. We can use an **inequality** to do this. We can start by finding an** inequality that includes x**. For example, we might have the inequality** x ≥ 0**. This inequality means that x is greater than or equal to 0.

Now, we can use this inequality to find a** lower bound for x**. We can do this by finding a number that is** less than or equal to 0**. One number that is less than or equal to 0 is –1. So, we can say that** -1 is a lower bound for x**.

There are two broad classes of lower bounds: (a) the** tight lower bound** and (b) the** loose lower bound**. The** tight lower bound** is the lower bound that is** closest to the actual value** of the function. The** loose lower bound** is the lower bound that is** farthest from the actual value** of the function.

## Explanation of Bounds with Visual Examples

All of the above discussion has been a **high level mathematical** stuff but in this section we will try to explain the concept with the help of **elementary level examples** and diagrams.

Lets **consider** a **set A** given below:

**A = {1, 3, 2, 5, 10, 11, 33, 15}**

Now there are **several methods** of finding the lower and upper bounds.

**First method** is to use a **conditional sub-division** or tree method as explained in the figure 1 above. Here we have simply assumed a **middle value** and **successively divided** the set into two equal parts until and unless only **one** of the **elements** remains in the set. It can be seen that the lowest value node represents the **lower bound** while the highest node represents the **upper bound.**

In practice, since the data is usually **very large,** so we normally prefer other **analytical solutions** of finding the **minima** and **maxima.** One of the methods is to plot the data in a **histogram plot** to visualize the bounds. As shown in the following figure, we can simply plot all the values in some bins, and find the highest and lowest value bin containing non-zero value:

**Figure 2:** Bounds using **Histogram Plot**

For the case of a **continuous function** such a depiction can also be made as shown in the **figure** below:

**Figure** **3:** Bounds on a **Continuous Function**

## Numerical Problem

Find the **upper** and **lower bound** on the following set of data points:

**B = {10, 20, 30, 50, 15, 110, 33, 25}**

### Solution

Here it can be seen that if we **successively divide** this set into smaller parts as shown in the figure below:

**Figure 4:** Upper and Lower Bounds on **Given Data**

As shown in the **figure,** we first divide the data on the **value 30** which leaves** 5 data points**. Then we split it on the **value 15** which leaves **2 data points** in the lower bound. Then we further split on 10 to get the final **lower bound**.

Similarly as shown in the **figure,** we first divide the data on the **value 30** which leaves 4** data points**. Then we split it on the **value 50** which leaves **2 data points** in the upper bound. Then we further split on 110 to get the final **upper bound.**

We can find that:

**Lower Bound = 10**

**Upper Bound = 110**

*All mathematical drawings and images were created with GeoGebra.*