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-
One-to-OneBack to Top
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.
OntoBack to Top
Now we will see types of function which are given below. There are three types of function.
One to one function
Many one function.
Here we will discuss onto function. Let’s talk about onto definition.
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:
For example: Suppose we have a onto function f (x) = 3x – 4, where f : R → 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.
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.
DomainBack to Top
Now we will understand how to calculate the domain in maths? Assume that we have a function and we want to find the domain of function.
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, -93), (-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.
Now we will understand that how to calculate domain in Math. 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.
Types of Functions and their GraphsBack to Top
One – one function
Many one function
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 → 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 → L is one – one if value of 'k' not equals to 'l'. (k ᚌl) → f (l) ᚌf (l) for all kl Ԑ K.
Many one function: - A function f: K → 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 → L is a many one function if there exist a, b Ԑ K such that a ᚌ b but f (a) = f (b).
Onto function: - Function f: K → 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:
This is all about types of functions and its graphs.