Is a function which is both an injection and Surjection in other words if every element of the codomain is the image of exactly one element from the domain?

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Índice Show

Is a function which is both an injection and surjection?
What is injectivity and Surjectivity of a function?
Which function is called surjective function?
When can we say that a function is injective or one

Surjections, Injections, and Bijections
Inverse Image

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A function is a rule that assigns each input exactly one output. We call the output the image of the input. The set of all inputs for a function is called the domain. The set of all allowable outputs is called the codomain. We would write $f:X \to Y$ to describe a function with name $f\text{,}$ domain $X$ and codomain $Y\text{.}$ This does not tell us which function $f$ is though. To define the function, we must describe the rule. This is often done by giving a formula to compute the output for any input (although this is certainly not the only way to describe the rule).

For example, consider the function $f:\N \to \N$ defined by $f(x) = x^2 + 3\text{.}$ Here the domain and codomain are the same set (the natural numbers). The rule is: take your input, multiply it by itself and add 3. This works because we can apply this rule to every natural number (every element of the domain) and the result is always a natural number (an element of the codomain). Notice though that not every natural number actually is an output (there is no way to get 0, 1, 2, 5, etc.). The set of natural numbers that are actually outputs is called the range of the function (in this case, the range is $\{3, 4, 7, 12, 19, 28, \ldots\}\text{,}$ all the natural numbers that are 3 more than a perfect square).

The key thing that makes a rule actually a function is that there is exactly one output for each input. That is, it is important that the rule be a good rule. What output do we assign to the input 7? There can only be one answer for any particular function.

The description of the rule can vary greatly. We might just give a list of the images of each input. You could also describe the function with a table or a graph or in words.

Example $\PageIndex{1}$

The following are all examples of functions:

$f:\Z \to \Z$ defined by $f(n) = 3n\text{.}$ The domain and codomain are both the set of integers. However, the range is only the set of integer multiples of 3.
$g: \{1,2,3\} \to \{a,b,c\}$ defined by $g(1) = c\text{,}$ $g(2) = a$ and $g(3) = a\text{.}$ The domain is the set $\{1,2,3\}\text{,}$ the codomain is the set $\{a,b,c\}$ and the range is the set $\{a,c\}\text{.}$ Note that $g(2)$ and $g(3)$ are the same element of the codomain. This is okay since each element in the domain still has only one output.
$h:\{1,2,3\} \to \{1,2,3\}$ defined as follows:

This means that the function $f$ sends 1 to 2, 2 to 1 and 3 to 3: just follow the arrows.

The arrow diagram used to define the function above can be very helpful in visualizing functions. We will often be working with functions with finite domains, so this kind of picture is often more useful than a traditional graph of a function. A graph of the function in example 3 above would look like this:

It would be absolutely WRONG to connect the dots or try to fit them to some curve. There are only three elements in the domain. A curve suggests that the domain contains an entire interval of real numbers. Remember, we are not in calculus any more!

Since we will so often use functions with small domains and codomains, let's adopt some notation that is a little easier to work with than that of examples 2 and 3 above. All we need is some clear way of denoting the image of each element in the domain. In fact, writing a table of values would work perfectly:

$x$01234$f(x)$33241

We simplify this further by writing this as a matrix with each input directly over its output:

\begin{equation*} f = \begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 7 & 7 & 7 & 7 & 7\end{pmatrix}. \end{equation*}

Is a function which is both an injection and surjection?

The function is bijective (one-to-one and onto, one-to-one correspondence, or invertible) if each element of the codomain is mapped to by exactly one element of the domain. That is, the function is both injective and surjective. A bijective function is also called a bijection.

What is injectivity and Surjectivity of a function?

An injective function is one in which each element of Y is transferred to at most one element of X. Surjective is a function that maps each element of Y to some (i.e., at least one) element of X.

Which function is called surjective function?

In mathematics, a surjective function (also known as surjection, or onto function) is a function f that every element y can be mapped from element x so that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of its domain.

When can we say that a function is injective or one

If the codomain of a function is also its range, then the function is onto or surjective. If a function does not map two different elements in the domain to the same element in the range, it is one-to-one or injective.