Injection, surjection, and bijection

In summary, injective, surjective, and bijective maps all have different properties that go in both directions.
  • #1
Koshi
18
0
I'm having trouble understanding just what is the difference between the three types of maps: injective, surjective, and bijective maps. I understand it has something to do with the values, for example if we have T(x): X -> Y, that the values in X are all in Y or that some of them are in Y...
Honestly I'm just incredibly confused about the terms. If someone could give me a straightforward way of explaining each of them I would very much appreciate it.
 
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  • #3
Remember the definition of a function f : X --> Y. It must satisfy two essential conditions:

1. Every element of X gets mapped to something in Y.
2. That something in Y is unique for each element of X.

Injections and surjections are special kinds of functions that also have one of these properties going in the other direction:

(Surj.) Every element of Y is mapped to by some element of X.
(Inj.) The element of X that maps to a particular value in Y is unique.

A function which is both surjective and injective is called bijective.
 
  • #4
Moo Of Doom said:
Remember the definition of a function f : X --> Y. It must satisfy two essential conditions:

1. Every element of X gets mapped to something in Y.
2. That something in Y is unique for each element of X.

Injections and surjections are special kinds of functions that also have one of these properties going in the other direction:

(Surj.) Every element of Y is mapped to by some element of X.
(Inj.) The element of X that maps to a particular value in Y is unique.

A function which is both surjective and injective is called bijective.

Wow, thank you so much! That was exactly the explanation I was looking for.
This will make my linear class so much easier to follow
 
  • #5
Glad to have been of help. :)
 

Related to Injection, surjection, and bijection

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

An injection is a function that maps distinct elements of the domain to distinct elements in the range. A surjection is a function that maps every element in the range to at least one element in the domain. A bijection is a function that is both injective and surjective, meaning it maps distinct elements of the domain to distinct elements in the range and every element in the range is mapped to by exactly one element in the domain.

2. Why are injections, surjections, and bijections important in mathematics?

Injections, surjections, and bijections are important because they help us understand the relationship between different sets and how they are connected. They also allow us to manipulate and transform sets in a meaningful way, which is essential in many areas of mathematics, such as algebra, calculus, and topology.

3. How can I determine if a function is injective, surjective, or bijective?

To determine if a function is injective, you can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not injective. To determine if a function is surjective, you can use the vertical line test. If a vertical line intersects the graph of the function at every point, then the function is surjective. To determine if a function is bijective, you can check if it passes both the horizontal and vertical line tests.

4. What is the inverse of a bijective function?

The inverse of a bijective function is a function that "undoes" the original function. It switches the inputs and outputs of the original function, such that the domain and range are swapped. This means that the inverse of a bijective function is also bijective.

5. Can a function be both injective and surjective, but not bijective?

No, a function cannot be both injective and surjective without being bijective. In order for a function to be bijective, it must be both injective and surjective. If a function is only injective, it means that there may be elements in the range that are not mapped to by any element in the domain. If a function is only surjective, it means that there may be elements in the domain that are not mapped to by any element in the range. A bijective function ensures that every element in the range is mapped to by exactly one element in the domain, and vice versa.

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