Is f Injective? Understanding the Composition of Functions

In summary, the function f is injective but not onto because there are two elements in the domain that map to the same element in the codomain.
  • #1
UOAMCBURGER
31
1

Homework Statement


Let A, B, C be finite sets such that A and B have the same number of elements, that is, |A| = |B|. Let f : A → B and g : B → C be functions.
(a) Suppose f is one-to-one. Show that f is onto.
(b) Suppose g ◦ f is one-to-one. Show that g is one-to-one.

Homework Equations

The Attempt at a Solution


We can let n = |A| = |B|
Since f is injective, we know that for all b ∈ B there exists at most ONE a ∈ A, such that f(a)=b right? that is, given x,y ∈ A f(x) != f(y) => x != y right? I can just see this from the definition, not sure how useful it is or where to go from here.
 
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  • #2
UOAMCBURGER said:

Homework Statement


Let A, B, C be finite sets such that A and B have the same number of elements, that is, |A| = |B|. Let f : A → B and g : B → C be functions.
(a) Suppose f is one-to-one. Show that f is onto.
(b) Suppose g ◦ f is one-to-one. Show that g is one-to-one.

Homework Equations

The Attempt at a Solution


We can let n = |A| = |B|
Since f is injective, we know that for all b ∈ B there exists at most ONE a ∈ A, such that f(a)=b right?
Yes.
... that is, given x,y ∈ A f(x) != f(y) => x != y right?
No. It means ##x\neq y \Rightarrow f(x) \neq f(y)## or ##f(x)=f(y) \Rightarrow x=y ##
... I can just see this from the definition, not sure how useful it is or where to go from here.
You have to use the finiteness of your sets. Injective means that all elements of ##A## are sent to different elements in ##B##. Now you have to show, that you get all of them.
 
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  • #3
fresh_42 said:
Yes.No. It means ##x\neq y \Rightarrow f(x) \neq f(y)## or ##f(x)=f(y) \Rightarrow x=y ##
You have to use the finiteness of your sets. Injective means that all elements of ##A## are sent to different elements in ##B##. Now you have to show, that you get all of them.
I think my definition was correct, I meant given two elements in the domain (hence x,y ∈ A) then f(x) != f(y) (meaning that no two elements in A can map to the same element in the codomain, B) .. the definition of injective functions. the x !=y just means that the two elements in the domain can't be equal, otherwise it is possible to have f(x) = f(y). sorry if i wasnt clear about that.
 
  • #4
UOAMCBURGER said:
I think my definition was correct,
No, it was not.
... I meant given two elements in the domain (hence x,y ∈ A)...
They are only different if you add ##x\neq y##
then f(x) != f(y) (meaning that no two elements in A can map to the same element in the codomain, B)
Yes, but that is not what you wrote. No two elements map onto the same element means: If two elements (##x, y##) happen to map to the same element (##f(x)=f(y)##), then they are equal (##x=y##):
$$
f \text{ is injective } \Longleftrightarrow (f(x)=f(y) \Longrightarrow x=y) \Longleftrightarrow (x \neq y \Longrightarrow f(x)\neq f(y))
$$
However, you wrote
UOAMCBURGER said:
that is, given x,y ∈ A f(x) != f(y) => x != y right?
which is the wrong direction and means that ##f## is well-defined: Different image points cannot result from one origin.
.. the definition of injective functions. the x !=y just means that the two elements in the domain can't be equal, otherwise it is possible to have f(x) = f(y). sorry if i wasnt clear about that.

I like the following mnemonic: Put your arms in front of you and let the hands be ##f##. Now make them touch at the wrist. This is not allowed for any function to be well-defined. Let them touch at the finger tips instead, then this is not allowed for injectivity.
 
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1. What is the definition of composition of functions?

Composition of functions is an operation that combines two or more functions to create a new function. The output of one function becomes the input of the next function, resulting in a chain of functions.

2. How is composition of functions written mathematically?

The composition of two functions f and g is written as (f ∘ g)(x), which means that the output of g is used as the input of f. This is read as "f composed with g of x."

3. What is the importance of the order of functions in composition?

The order of functions in composition is crucial because it determines the resulting function. Changing the order of functions may result in a completely different function with different outputs.

4. Are there any restrictions on the types of functions that can be composed?

Yes, there are some restrictions on the types of functions that can be composed. Both functions must have the same number of inputs and outputs, and the output of the first function must match the input of the second function.

5. How is composition of functions used in real life?

Composition of functions is used in various fields such as economics, engineering, and computer science. It is used to model complex systems, analyze data, and create efficient algorithms.

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