Injection, Surjection, Bijection

In summary, The given function f(n) = n+1 is an injection but not a surjection. An injection means that each element in the codomain has at most one original, while a surjection means that each element in the codomain has at least one original. This can be seen by examining the definitions and the given function.
  • #1
KOO
19
0
Can anyone explain to me how to do these types of questions? I have the answers but I don't understand it.

The function f: N -> N, f(n) = n+1 is
(a) Surjection but not an injection
(B) Injection but not a surjection
(c) A Bijection
(d) Neither surjection not injection

The answer is B: Injection but not a surjection

How to do these types of questions?
 
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  • #2
KOO said:
Can anyone explain to me how to do these types of questions? I have the answers but I don't understand it.

The function f: N -> N, f(n) = n+1 is
(a) Surjection but not an injection
(B) Injection but not a surjection
(c) A Bijection
(d) Neither surjection not injection

The answer is B: Injection but not a surjection

How to do these types of questions?

We start with the definitions.
A function is surjective if each element in the codomain has at least 1 original.
A function is injective if each element in the codomain has at most 1 original.
A function is bijective if it is both surjective and injective.

If we pick some element in the codomain N, how many originals does it have?

Btw, the answer is not B. Where did you get that from?
 
  • #3
KOO said:
Can anyone explain to me how to do these types of questions? I have the answers but I don't understand it.
The function f: N -> N, f(n) = n+1 is
(a) Surjection but not an injection
(B) Injection but not a surjection
(c) A Bijection
(d) Neither surjection not injection
The answer is B: Injection but not a surjection
How to do these types of questions?
It is easy if one learns the definitions.

Any function is a set of ordered pairs,
This function is \(\displaystyle f=\{(n,n+1) :n\in\mathbb{N}\}\).

A function is injective if no two pairs have the same second terms.

A function is surjective if every term if the finial set is the second term of some pair.
 

1. What is the difference between injection, surjection, and bijection?

Injection, surjection, and bijection are terms used in mathematics to describe the relationship between two sets. An injection is a function where each element in the domain maps to a unique element in the range. A surjection is a function where each element in the range is mapped to by at least one element in the domain. A bijection is a function that is both injective and surjective, meaning each element in the domain maps to a unique element in the range and each element in the range is mapped to by exactly one element in the domain.

2. How can you determine if a function is injective, surjective, or bijective?

To determine if a function is injective, we can use the "horizontal line test." This means that for each y-value in the range, there should only be one x-value in the domain that maps to it. To determine if a function is surjective, we can use the "vertical line test." This means that for each x-value in the domain, there should be at least one y-value in the range that is mapped to it. To determine if a function is bijective, we can use both the horizontal and vertical line tests.

3. What is an example of a function that is injective, surjective, and bijective?

An example of a function that is injective, surjective, and bijective is f(x) = x. This function maps each element in the domain to itself, making it both injective and surjective. Therefore, it is also bijective.

4. Can a function be injective but not surjective?

Yes, a function can be injective but not surjective. This means that each element in the domain maps to a unique element in the range, but not every element in the range is mapped to by at least one element in the domain. An example of this is the function f(x) = x^2, where the range is all positive numbers and does not include 0.

5. Why are injection, surjection, and bijection important in mathematics?

Injection, surjection, and bijection are important concepts in mathematics because they help us understand the relationship between two sets. They are especially useful in the study of functions and their properties. For example, bijections are often used in proofs to show that two sets have the same cardinality (number of elements). These concepts also have applications in other fields, such as computer science and economics.

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