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Functions in Discrete Mathematics

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A function in discrete Math named ‘$g$’ from the Set ‘$P$’ to ‘$Q$’ is a relation from $P$ to $Q$ that satisfies the below conditions

For each value of ‘$a$’ belonging to ‘$P$’, there will exist a unique value ‘$b$’ which belongs to ‘$Q$’ with both the values of ‘$a$’ and ‘$b$’ belonging to the function ‘$g$’. The set ‘$P$’ is the Domain of the function ‘$g$’ and ‘$Q$’ is the co domain or range of the function ‘$g$’. A function ‘$g$’ from $P$ to $Q$ is denoted as $g$: $P \rightarrow Q$.

Suppose that ‘$P$’ and ‘$Q$’ are two non empty Sets then the function $g$: $P \rightarrow Q $is a one to one or can be named as injective function if no two elements of ‘$P$’ are mapped to the same element of ‘$Q$’ that is for ‘$a$’ and ‘$b$’ belongs to ‘$P$’ and the Functions $g (a)$ and $g (b)$ belongs to ‘$Q$’, with the condition that a is not equal to ‘$b$’ that implies the result that $g (a)$ also not equal to $g (b)$.

A Onto function is there for every ‘$b$’ which belongs to ‘$Q$’ satisfy the condition $g (a)$ = $g (b)$. A function will be bijective if it is both one- one and onto. The real values in discrete mathematics which are assigned to each number ‘$x$’ belongs to ‘$R$’ for a particular value $y = g (x)$ where ‘$y$’ also belongs to ‘$R$’. 
Basically there are three types of functions as listed below;
One to one function
Onto function
Many one function.

One-to-One

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A function (denoted as $f$) from $P$ to $Q$ is said to be one – to – one function, if and only if $f\ (P)$ = $f\ (Q)$ then $P$ = $Q$. In other words every element in $Q$ has only one pre image in element $P$. One – to – one function diagram is shown below:

One to One Function
 
According to definition of One to One Function we can say that element of $Q$ has one pre image in element $P$.

Now we will see One to One correspondence.

One – to – one correspondence is defined as a situation such that elements of one Set (let a set $U$) can be properly (evenly) matched with elements of second set (other set $V$). Here meaning of word evenly is every element of set '$U$' corresponds to one and only one member of set '$V$' and each element of set '$V$' corresponds to one and only one member of set '$U$'. So we can say that each element of set '$U$' is paired with exactly one element of set '$V$' and vice versa. If we see the terms of Ordered Pair $(x,\ y)$ here '$x$' shows the element of set '$U$' and '$y$' is an element of set '$V$'. Two orders are not possible for this matching which has first element same, two orders are not possible for same element. If this type of matching exists in a set than it shows one – to – one correspondence between Sets $U$ and $V$.

We can say that if two sets have same Cardinality than one – to – one correspondence exists among two sets. Now we will talk about one - to - one function. Commonly one - to – one function is used to check whether one – to – one correspondence exists among infinite sets. This is all about one to one correspondence and Functions.
A function (denoted by $f$) from $P$ to $Q$, is said to be onto function if and only if for all values of $q$ in $Q$ there is an $p$ in $P$ such that $f\ (p)$ = $q$. We can say that all elements in set $Q$ are conserved. Now we will understand the concept of onto function with the help of example.

Example: 

Suppose we have a onto function $f\ (x)$ = $3x - 4$, where $f : R \rightarrow R$.

Solution: 

Here Given function is $f\ (x)$ = $3x - 4$, when we solve the given function then we get a straight line that is shown in graph mentioned below.

Onto Function Graph

If we move further to solve the function then every possible '$y$' value can be used. Straight line we get possesses or follows the property in which every $x$ – value has unique $y$ – value which is not used by any of the other $x$ – values. This is all about onto function.

Domain

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Function displays the relationship between values. Every input term of a function gives back one output value. It is basically written as $f\ (p)$ where ‘$p$’ is value you assign to it.

Do you know how to calculate the domain in maths? Let have a brief description.
To calculate the domain of a function we need to follow some steps which are as follows:

Step 1: First we have to assume a function which contains ‘$x$’ and ‘$y$’ coordinates.

Step 2: Using definition of domain of function, domain of a function is all ‘$x$’ coordinate values.

In this way we can calculate domain of a function and this process is called as domain math.
Assume that we have a function and we want to find the domain.

All ‘$x$’ coordinate values of a function are domain of a function. In the same way, all possible ‘$y$’ coordinate values are range of a function.
Suppose we have some values $(44, -51),\ (-38, 54),\ (86, -89),\ (-15, 17),$ then domain of function is all ‘$x$’ coordinate values.

Domain = $44,\ -38,\ 86,\ -15$.

Range is all ‘$y$’ coordinate values,

Range = $-51,\ 54,\ -89,\ 17$.

Types of Functions and their Graphs

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Function can be used to represent relation between Set of inputs values and set of outputs values in such a way that every value of input is related to exactly one value of output.

Let’s discuss the types of function and its graph in detail.
One – one function (it is also called as injection function): - Function $f: K \rightarrow L$ is called as one – one function if every input term of element '$K$' has different image in '$L$'. So it can be written as: $f:\ K \rightarrow L$ is one – one if value of '$k$' not equals to '$l$'. $(k \neq l) \rightarrow f (l) \neq f (l)$ for all $kl \epsilon K$.

One One Function
Many One Function: A function $f:\ K \rightarrow L$ is called as many one function if two or more elements of set '$K$' have same images in '$L$'. In mathematical form it can be written as:

$f:\ K \rightarrow L$ is a many one function if there exist $a, b \epsilon K$ such that $a \neq b$ but $f\ (a)$ = $f\ (b)$.

Many One Function
Onto Function: Function $f:\ K \rightarrow L$ is called as Onto function or it is also called as 'surjection' if every value of element '$L$' is image of some element of '$K$' i.e. if $f\ (K)$ = $L$, and range of '$f$' is co – domain of function '$f$' or in other words we can say that elements of '$L$' has no pre – image in element '$K$'. In mathematical form it can be written as:

Onto Function