What is Onto Function? How to Prove a Function is Onto? 

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Master the concept of Onto Functions and sharpen your skills in proving the surjectivity of functions. Our comprehensive resource guides you through the process, equipping you with the tools to validate that a function covers its entire target space.

What is Onto Function?

An onto function, also known as a surjective function, is a mathematical concept that describes the relationship between two sets. It essentially means that every element in the target set (codomain) has at least one corresponding element in the starting set (domain).

Here’s a breakdown of the key points:

Definition: A function f: A → B is onto if for every element b in B, there exists at least one element a in A such that f(a) = b.

In simpler terms: The function “maps” all the elements in the target set even if some elements are mapped to multiple elements in the starting set.

Some key characteristics:

• The range of the onto function is equal to the codomain.
• There may be elements in the starting set that are not mapped to any element in the target set.
• The function’s graph intersects every horizontal line at least once.

Examples of onto functions:

• f(x) = 2x, where A = all real numbers and B = all non-negative real numbers.
• f(x) = x^2, where A = all non-negative real numbers and B = all non-negative real numbers.
• f(x) = x + 1, where A and B are both the set of all integers.

Examples of non-onto functions:

• f(x) = x^2, where A and B are both the set of all real numbers.
• f(x) = sin(x), where A and B are both the set of all real numbers.

Here are some additional ways to understand onto functions:

• Imagine throwing a ball at a target. If the target is big enough that every throw lands on the target, then the throwing motion is an onto function.
• Think of a function as a machine that takes inputs and produces outputs. If the machine can produce every possible output, then it is an onto function.

Number of Onto Functions

The number of onto functions between two sets depends on the size of the sets. Here’s how to calculate it:

Formula:

If set A has m elements and set B has n elements, the number of onto functions from A to B is:

nm – nC₁(n-1)m + nC₂(n-2)m – nC₃(n-3)m + … – nCn-₁ (1)m

where:

• nm is the total number of functions from A to B,
• nC₁ is the number of combinations of (n-1) elements from n,
• nC₂ is the number of combinations of (n-2) elements from n,
• … continues similarly for all combinations up to (n-n+1) elements,
• (1)m is 1 multiplied by itself m times.

Explanation:

1. nm: This represents the total number of functions from A to B, where each element in A can be mapped to any of the n elements in B.
2. nC₁(n-1)m: This subtracts the number of functions that miss exactly one element in B. We choose (n-1) elements from n to miss and multiply by m for the number of ways to map the remaining element to any element in B.
3. nC₂(n-2)m: This further subtracts the number of functions that miss exactly two elements in B. We choose (n-2) elements from n to miss and multiply by m for the remaining elements.
4. The process continues for all possible combinations of missing elements, until we reach nCn-₁ (1)m, which accounts for the functions that miss only one element.

Example:

Let set A have 3 elements and set B have 4 elements.

• Number of onto functions = 3*4 – 4C1(4-1)3 + 4C2(4-2)3
• = 12 – 427 + 69
• = 12 – 108 + 54
• = 54

Therefore, there are 54 onto functions from A to B.

Note:

• This formula applies when m ≥ n. For m < n, there are no onto functions.
• This formula is also known as the Inclusion-Exclusion Principle.

Onto Function Formula

The formula for the number of onto functions (also known as surjective functions) from a set A with m elements to a set B with n elements is:

nm – nC1(n-1)m + nC2(n-2)m – nC3(n-3)m + … – nCn-1(1)m

where:

• nm is the total number of functions from A to B (m^n).
• nC1(n-1)m is the number of functions that map exactly one element of A to each element of B.
• nC2(n-2)m is the number of functions that map exactly two elements of A to each element of B.
• … continues this pattern for all possible ways of mapping elements of A to B.
• nCn-1(1)m is the number of functions that map all elements of A to the same element of B.

Note: This formula only works when m >= n. If m < n, there are no onto functions from A to B.

Here are some additional points to remember about the formula:

• The formula is based on the principle of inclusion-exclusion, which is a counting technique used to count the number of objects that satisfy certain conditions.
• The formula can be difficult to calculate by hand for large values of m and n. However, there are many online calculators and computer programs that can calculate the number of onto functions for you.
• There are also other formulas that can be used to calculate the number of onto functions, such as the Stirling numbers of the second kind.

Here are some examples of how to use the formula:

Example 1: How many onto functions are there from the set A = {1, 2, 3} to the set B = {a, b}?

In this case, m = 3 and n = 2. Therefore, the number of onto functions is:

3^2 – 2C1(2-1)3 + 2C2(2-2)3 = 9 – 2 * 3 + 0 = 3

Example 2: How many onto functions are there from the set A = {1, 2, 3, 4} to the set B = {a, b, c}?

In this case, m = 4 and n = 3. Therefore, the number of onto functions is:

4^3 – 3C1(3-1)4 + 3C2(3-2)4 – 3C3(3-3)4 = 64 – 3 * 4^2 + 3 * 4 – 0 = 16

How to Prove a Function is Onto?

A function is said to be onto if every element in its codomain is the image of at least one element in its domain. In other words, for every element “y” in the codomain, there exists at least one element “x” in the domain such that f(x) = y. Here’s how you can prove a function is onto:

Direct Proof:

1. State the function and its domain and codomain: Clearly define the function f, its domain A, and its codomain B.
2. Choose an arbitrary element from the codomain: Let “y” be any element in B.
3. Show that there exists an element in the domain that maps to “y”: This is the crucial step. You need to demonstrate that there exists an element “x” in A such that f(x) = y. You can achieve this by:
   • Solving the equation f(x) = y for x: If you can express x explicitly in terms of y, then you’ve proven the existence of such an element.
   • Providing a specific element “x” in A: If you can find a specific element in A that maps to “y” through f, then you’ve shown that f is onto.
   • Using logic or set theory: In some cases, you might be able to utilize logical arguments or properties of sets to prove that for every “y” in B, there exists a corresponding “x” in A.
4. Repeat step 2 for all elements in the codomain: Since you’ve chosen “y” arbitrarily, you need to show that the same holds true for all elements in B.
5. Conclude: If you’ve successfully shown that for every element “y” in B, there exists an element “x” in A such that f(x) = y, then you’ve proven that the function f is onto.

Alternative Proof:

1. Show that the range of the function is equal to its codomain: The range of a function is the set of all its output values. If you can demonstrate that the range of f is exactly equal to B, then you’ve indirectly proven that f is onto.

Tips:

• When solving for x, consider the properties of the function and the sets involved.
• Look for patterns or special cases that can simplify the proof.
• Visualizing the function and its domain and codomain may be helpful in understanding the relationships between elements.
• Practice proving different types of functions to develop your understanding.

What is One One and Onto Function?

A function can be classified as one-to-one, onto, or both, depending on how it relates elements in its domain and range. Here’s a breakdown of each:

One-to-one (Injective):

• Every distinct element in the domain maps to a distinct element in the range.
• No two different elements in the domain can have the same image under the function.
• Graphically, no horizontal line intersects the graph of the function more than once.

Onto (Surjective):

• Every element in the range is the image of at least one element in the domain.
• All elements in the range are “used” by the function.
• Graphically, the range of the function covers the entire target set.

One-to-one and Onto (Bijective):

• The function satisfies both the one-to-one and onto properties.
• Every element in the domain has a unique image in the range, and all elements in the range have a pre-image in the domain.
• Such functions are invertible, meaning you can find a unique function that maps elements back from the range to the domain.

Here are some examples to illustrate these concepts:

One-to-one:

• f(x) = x + 1 (maps every x to a unique x+1)
• f(x) = 2x (maps every x to a unique 2x)

Onto:

• f(x) = x^2 (maps every x to some x^2 value, but not all x^2 values)
• f(x) = 2x + 1 (maps every x to some 2x+1 value, covering all possible values)

Bijective:

• f(x) = x (maps every x to a unique x)
• f(x) = x^3 (maps every x to a unique x^3, and every x^3 is the image of some x)

Understanding these classifications helps analyze and solve problems related to functions and their properties. It also plays a crucial role in various mathematical areas like linear algebra and group theory.

Properties of Onto Function

An onto function, also known as a surjective function, has several key properties. Here are some of the most important ones:

1. Every element in the codomain has at least one preimage.

This means that for every element in the output set (codomain), there exists at least one element in the input set (domain) that maps to it. In other words, the function “hits” every element in the output set.

2. The range of the function is equal to the codomain.

This follows from the definition of an onto function. Since every element in the codomain has at least one preimage, all elements in the range (the set of all outputs of the function) must be present in the codomain.

3. Every onto function has a right inverse.

A right inverse is a function that “undoes” the original function. That is, if f(a) = b, then the right inverse g(b) = a. Not all functions have right inverses, but onto functions do.

4. Every function that has a right inverse is an onto function.

This is the converse of the previous property. If a function has a right inverse, then it must be onto.

5. Onto functions are not necessarily one-to-one.

A one-to-one function, also known as an injective function, is a function that maps each element in the domain to a unique element in the codomain. An onto function can be both one-to-one (in which case it is called a bijective function) or not one-to-one.

6. The graph of an onto function intersects every horizontal line at least once.

This can be used as a visual aid to determine if a function is onto. If the graph of the function intersects every horizontal line at least once, then the function is onto.

7. The cardinality of the domain is greater than or equal to the cardinality of the codomain.

Cardinality refers to the number of elements in a set. This property follows from the fact that every element in the codomain must have at least one preimage in the domain.

Here are some additional notes about onto functions:

• The term “surjective” is often used interchangeably with “onto.”
• The properties of onto functions are important in various branches of mathematics, including set theory, algebra, and analysis.
• Onto functions play a crucial role in several real-world applications, such as computer science, engineering, and economics.

Solved Examples on Onto Function

Here are a few solved examples to help you understand the concept of an onto function:

Example 1:

Function: f(x) = x^2 Domain: {all real numbers} Codomain: {all real numbers greater than or equal to 0} Question: Is f(x) an onto function?

Solution:

An onto function maps every element in its domain to some element in its codomain. In this case, for any non-negative real number y in the codomain, we can find a real number x in the domain such that x^2 = y. Therefore, f(x) is an onto function.

Example 2:

Function: g(x) = 2x + 1 Domain: {all real numbers} Codomain: {all real numbers} Question: Is g(x) an onto function?

Solution:

For any real number y, we can find a real number x such that g(x) = y. Specifically, x = (y – 1) / 2 satisfies this condition. Therefore, g(x) is an onto function.

Example 3:

Function: h(x) = |x| Domain: {all real numbers} Codomain: {all non-negative real numbers} Question: Is h(x) an onto function?

Solution:

For any non-negative real number y, we can find a real number x such that h(x) = y. Specifically, x = y or x = -y satisfy this condition. However, for negative real numbers, there is no real number x such that h(x) = y. Therefore, h(x) is not an onto function.

Example 4:

Function: f(x) = x + 1 Domain: {1, 2, 3, 4} Codomain: {2, 3, 4, 5} Question: Is f(x) an onto function?

Solution:

Yes, f(x) is an onto function. Every element in the codomain can be reached by applying the function to one or more elements in the domain.

Example 5:

Function: g(x) = x^2 Domain: {1, 2, 3, 4} Codomain: {1, 4, 9, 16} Question: Is g(x) an onto function?

Solution:

No, g(x) is not an onto function. The element 2 in the codomain cannot be reached by applying the function to any element in the domain.

These are just a few examples to help you understand onto functions. There are many other functions that can be classified as onto or not onto. The key is to remember that an onto function maps every element in its domain to some element in its codomain.

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