Onto Function: One to One, Bijective and Surjective Function, Examples

Onto Function is also known as Surjective Function. A function is defined as that which relates values/elements of one set to the values/elements of a different set, in a way that elements from the second set is equivalently defined by the elements from the first set. A function comprises various types which usually define the relationship between two sets that are in a different pattern. The different types of function are: onto function, many to one function, one to one function, etc.

Key Terms: Onto Function, total number of functions, right inverse, surjective functions, injective function.

Read More: Formulas of Inverse Trigonometry

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What is Surjective/Onto Function?

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The function is a process that relates elements of two different sets. One set is called the domain while the other is referred to as codomain. Functions can be explained in simple words as ‘for every set of inputs, there is a unique set of outputs’. It is important to note that the output and input sets should have some elements for a function to exist, which means it must not be empty sets. 

Onto Function

Representation of a function is generally done as f (x) = y. There are different types of functions like identical functions, periodic functions, many to one functions, algebraic functions, onto function, into the function, rational functions, one to one function, linear, quadratic and cubic functions, even and odd functions etc. 

Every element of a function in the domain has one and only one element in the codomain. Two elements in the codomain can’t be related to any element in the domain. However, the same element in the codomain can be correlated with two elements in the domain.

Read more: Binary Operations

Onto Function Explained

For Explaining the Onto Function we have to consider two sets. So, let us consider Set A and Set B consisting of elements. If at least one or more elements are matching with A for every element B, then the function is said to be surjective or onto function. Nicolas Bourbaki was the person who coined the term Surjective Function.

In Fig 1 we can observe that there is a pre-image for every element of B or a matching element in Set A. Therefore, it is a surjective function. In Fig 2, since 1 element in Set B is not linked with any element of set A, it's not a surjective function.

Read more: Derivative Inverse Trigonometry

Number of Surjective functions/Onto Functions Formula

When we are about to find the number of onto functions from a set A containing ‘n’ number of elements to set B which has ‘m’ number of elements.

Thus,

Number of functions from A to B = mn

Total number of surjective functions = Total number of functions – Number of functions which cannot be referred as onto

Read more: Types of Functions

The formula to find the total number of functions that can’t be onto is given by:

(m1)(m−1)n+(m2)(m−2)n−(m3)(m−3)n+.....−(mm−1)(1)n

Therefore, by this formula, we can find the number of Onto functions.

mn−(m1)(m−1)n+(m2)(m−2)n−(m3)(m−3)n+.....−(mm−1)(1)n

When,

n < m, the number of onto functions = 0

n = m, the number of onto functions = m!

Read more: Relations and Functions

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Properties of an Onto Function

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The range denotes the actual input of the function, the codomain states possible outcomes, the domain is basically what can go into the function.

• All elements in onto function have a right inverse.
• All function is said to be Surjective if it is with a right inverse.
• It results in onto function only if we compose onto functions.

Read more: Types of Relations

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Onto Function Solved Examples

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Example 1: Let A = {1, 5, 8, 9) and B {2, 4} And f={(1, 2), (5, 4), (8, 2), (9, 4)}. Then prove f is a surjective function.

Solution: From the question itself we get,

A={1, 5, 8, 9)

B{2, 4}

& f={(1, 2), (5, 4), (8, 2), (9, 4)}

So, we can say elements 1 and 8 & 5 and 9 have the same range 2 & 4 respectively, that is every element on set B has a domain element on set A.

Therefore, f: A → B is a surjective function.

Example 2: Find the number of onto functions from the set X = {1, 2, 3, 4} to the set y= {a, b, c} .

Solution: Given: Set X = {1, 2, 3, 4}; Set Y = {a, b, c}

Here, n=4 and m=3

Then, the values of m and n in the formula are substituted and we get

= 34 – 3C1(2)⁴ + 3C2(1)⁴

= 81 – 3 (16) + 3(1)

= 81 – 48 + 3

= 84 – 48

= 36

Therefore, the number of surjective functions from set X to set Y is 36.

Read more: Triangle

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Things to Remember

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• A function serves the purpose of relating elements of two different sets. 
• Every element of a function in the domain has one and only one element in the codomain.
• If at least one or more elements are matching with A for every element B, then the function is said to be surjective or onto function.

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