A **correspondence rule** consists in **assign a unique element from a certain set** to **each unique element of another** **set**. This concept is often used when working with **math functions**.

When defining a mathematical function, what you do is establish the **half** through which the correspondences between two sets must be made. The **function** in itself, therefore, it acts as a rule of correspondence. In other words, the calculation of a function consists of **find out what the general correspondence is** that exists in one set with respect to another.

## Match rule types

We can distinguish between two large classes of correspondence rules. The **one-to-one correspondence** implies that each **element** of the set known as **Domain** corresponds to a single element of one named **Codomain**. The **one-to-one correspondence**, for its part, assumes that the inverse correspondence is also univocal (that is, to each element of the **Codomain** corresponds to a single element of the **Domain**).

From these first basic definitions it can be deduced that for a correspondence to be one-to-one it must also be unique. On the other hand, it should be mentioned that not always an image corresponds to each of the elements of the first set, nor do those of the second **source**.

Thinking for a moment about the **theory** of the sets, the graphical representation of all the possible correspondences between two sets (domain and codomain) gives us two more: that of univocal correspondences (which we can call **TO**) and that of the biunivocals (**B**). When observing the latter in a Venn diagram (the classic way of representing sets graphically, usually with circles or ovals that enclose the elements of each set), it is clearly evident that **B** is a subset of **TO**.

## Example of correspondences

Let’s take a **set A**, which is formed by **3**, **4** Y **5**, and a **set B**, which is composed of **9**, **12** Y **fifteen**. The correspondence between the two is the **triple**. In this way, the correspondence rule allows linking each element **Domain** (the **set A**) to an element of the **Codomain** (the **set B**).

*f (x) = 3x*

f (3) = 3 × 3 = 9

f (4) = 3 × 4 = 12

f (5) = 3 × 5 = 15

Domain = {3,4,5}

Codomain = {9,12,15}

This correspondence rule can also be graphed. Each element must be included within its corresponding set (**3**, **4** Y **5** in the **set A** Y **9**, **12** Y **fifteen** in the **set B**) and then match each element with a **arrow** according to the rule of correspondence.

## Other kinds of matching rule

But the rules of correspondence are not limited to these two possibilities; for example, the **not unique** occurs when there is at least one element of the first set for which there are two **images** or more. The aforementioned example would not serve to understand this situation, since each number only corresponds to a triple; But, if we talk about a set of people and a codomain of countries, and we relate them according to the countries that each person has visited, it is likely that some have never traveled, that others have simply gone to only one and that the rest have known **more than one**.

The correspondence **univocal, not biunivocal**, for its part, is one in which each element of the domain corresponds to a single image, but this does not happen in **sense** contrary. If none of the people in the previous example have traveled to more than one country, but two or more of them have visited the same country, then that country **has two or more origins**.

When establishing a correspondence rule, we must take into account different elements and concepts. One of them is the **rank**, which defines the set of possible values for the dependent variable, that is, the one that depends on the one chosen in the domain.