# What is the Definition and Understanding of Surjective Functions?

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• Phys12
In summary, a surjective function is one in which every element in the output set has at least one corresponding input element. This allows for the possibility of multiple input elements mapping to the same output element. The function itself, however, cannot have more than one input element for a given output element.
Phys12
In my book, the definition of surjection is given as follows:

Let A and B be sets and f:A->B. The function f is said to be onto if, for each b ϵB, there is at least one a ϵ A for which f(a)=b. In other words, f is onto if R(f)=B. A function which is onto is also called a surjection or a surjective function.

However, what I don't understand is why does there need to be at least one a ϵ A? Shouldn't there be only one since it's a function and a function by definition, for a given image, cannot have 2 pre-images?

Phys12 said:
In my book, the definition of surjection is given as follows:

Let A and B be sets and f:A->B. The function f is said to be onto if, for each b ϵB, there is at least one a ϵ A for which f(a)=b. In other words, f is onto if R(f)=B. A function which is onto is also called a surjection or a surjective function.

However, what I don't understand is why does there need to be at least one a ϵ A? Shouldn't there be only one since it's a function and a function by definition, for a given image, cannot have 2 pre-images?
No. A function cannot have two ##b \in B## for the same ##a \in A##. It can, however, have two elements ##a## which map onto the same element ##b##. E.g. ##f\, : \,x \longmapsto x^2## is a function, and ##f(-1) = f(+1)##. The relation ##x \longmapsto \pm \sqrt{x}## is no function, only if we restrict ourselves to either ##+\sqrt{x}## or ##-\sqrt{x}##, but not both. ##f \, : \,\mathbb{R} \longrightarrow \mathbb{R}## with ##f(x)=x^2## is not surjective, because the range is only ##\mathbb{R}_0^+ \subsetneq \mathbb{R}##. But ## f \, : \, \mathbb{R} \longrightarrow \mathbb{R}_0^+## with ##f(x)=x^2## is surjective.

Phys12

## 1. What is a surjection?

A surjection is a mathematical function that maps elements from one set to another in such a way that every element in the second set has at least one corresponding element in the first set. In other words, every element in the second set is mapped to by at least one element in the first set.

## 2. How is a surjection different from an injection?

A surjection is different from an injection in that an injection is a function where every element in the second set is mapped to by at most one element in the first set. This means that an injection is a one-to-one mapping, while a surjection is a onto mapping.

## 3. What is the importance of surjections in mathematics?

In mathematics, surjections are important because they allow us to determine if a function is bijective, which means it is both an injection and a surjection. Bijective functions have an inverse, which is crucial in many mathematical concepts such as solving equations and proving theorems.

## 4. Can a surjection have more than one output for a given input?

No, a surjection cannot have more than one output for a given input. This would violate the definition of a function, which states that each input can only have one output. If a surjection has more than one output for a given input, it would not be a valid function.

## 5. How do you prove that a function is a surjection?

To prove that a function is a surjection, you must show that every element in the second set has at least one corresponding element in the first set. This can be done by either directly showing that every element in the second set is mapped to by at least one element in the first set, or by using the contrapositive of the definition of a surjection, which states that if an element in the second set is not mapped to by any element in the first set, then the function is not a surjection.

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