# Difference Between Function and Relationship

Its applications range from the most basic, such as enumeration, addition, subtraction, among others, whose use is commonly observed in everyday life; but mathematics is also required in specialized studies to develop the most complex activities in areas such as architecture, medicine, computing…

Some of its precepts result in confusing processes, due to the similarity between them, such is the case of function and relation, which refer to the correspondence between sets but each one from a different perspective.

## What is a Function?

In mathematics, a function is defined as the relationship between two magnitudes, provided that the value of one depends on the other. An example of this can be explained through an everyday situation, which is that the cooking time of a cake depends on the temperature of the oven. Both values ​​are variables and according to their position, they are considered as:

• Dependent variable: it is the one whose value varies depending on the other magnitude, in the previous example it would be the cooking time.
• Independent variable: it is the value that conditions the dependent variable. We are referring to the temperature of the oven, returning to the example.

The values ​​are grouped into two sets, where each magnitude of group A has a unique and exclusive correspondence with another measure of set B. The data of group A is called domain, because it is the element that functions as the independent variable; while the data of set B is made up of the data that serves as a dependent variable and is called a codomain.

According to the correspondence between the elements of a set, the functions can be classified into three types:

• Injective: occurs when different elements of the set A and B correspond without repeating.
• Surjective: or also known as subjective, and applies when each element of set A is linked to an image of group B, even if they are shared.
• Bijective: in this function the injective and the surjective are combined in the sets, and it is that each element is linked and there are no values ​​of the sets to be related.

It is worth noting that a function can be represented through equations or algorithms that reflect the dependency of the variables, through tables that link the corresponding values, or with graphs that show the behavior of the elements and their connection.

## What is a Relationship?

When we refer to a relationship from mathematics, we talk about the link that exists between two given sets, in which each element of one group corresponds to at least one value of the other group. When only a magnitude of the set A is linked to a value of B, we are referring to the functions, so these are also considered as mathematical relations, but the opposite is not the case, so the relations are not functions.

In developing a mathematical relationship, the first set is called the domain , while the second is called the range or range . An example of this could be seen in the supermarket where each product has a value, the domain being the product itself, while the range is the price it has for the purchase.

Mathematical relationships can be represented through graphic schemes called Cartesian planes, revealing the behavior of the relationships between the magnitudes contained in two groups.

The applicability of mathematical relations outside their area of ​​origin is varied, and it is that in everyday life, unconsciously, we make use of it and it is that on a daily basis we establish links between different groups since it helps us to organize ourselves and to be participants in different activities.

According to the number of sets present in the relation, it is classified into three types:

• Unary relation: occurs when there is only one set and from there a subset is derived with elements that belong to it.
• Binary relation: As its name implies, it refers to two sets that are linked in various ways and their subsets are grouped as ordered pairs.
• Ternary relationship: the elements are linked to three different sets, so we speak of one or more triples, which is equivalent to ordered pairs of three elements.

## Difference Between Function and Relationship

• A mathematical function occurs when two elements of two different sets are exclusively linked to each other. For its part in the relationship, there may be correspondence between a value of a group A, with one or more of group B.
• The mathematical relation is always a function, but the function cannot be seen as a relation.
• In the function and in the relation, the set A receives the domain name; however, B, the function calls it a codomain, while the relation calls it a range or range.
• According to the correspondence that exists between the terms, the function is classified as injective, surjective and bijective. As opposed to relation which sets its types as unary, binary, or ternary.
• For the function, there is a dependency between the values ​​of the sets. On the other hand, for the relation, any set of ordered pairs that have correspondence between their elements is considered.