Proof of Injective Function Property

In summary, the conversation discusses the proof that if a function is injective, then its image of the difference of two sets is a subset of the difference of the image of the two sets. The conversation also touches on the idea of adjusting the sets in order to prove the relation for all subsets. It is concluded that if any element of the image of the difference of two sets is also in the image of one of the sets, then the function is not injective.
  • #1
Kamataat
137
0
I had this question on a test today.
Prove that if a function f:X-->Y is injective, then [itex]f(X\setminus A) \subset Y\setminus f(A), \forall A \subset X[/itex].

This is how I did it:
If x_1 is in A, then y_1=f(x_1) is in f(A). Because the function is injective, we can pick (cut Y into pieces) f(A) and f(X\A) so, that their intersection is empty (adjust A if needed). So, if x_2 is not in A = if x_2 is in X\A, then y_2=f(x_2) in f(X\A). From this we get that y_2 is in Y\f(A), since Y\f(A)=f(X\A). So, I've shown using the injective property, that y_2 in Y\f(A) follows from y_2 in f(X\A).

Right? Wrong?

- Kamataat
 
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  • #2
At first glance, it appears to be wrong. You say that you might adjust A, but that's not "legal". You were asked to prove the given relation for /all/ subsets A of X.

It should be clear that any element of f(X\A) is in Y. To prove that any element of f(X\A) is not in f(A), one might try to derive a contradiction...
 
  • #3
Muzza said:
At first glance, it appears to be wrong. You say that you might adjust A, but that's not "legal". You were asked to prove the given relation for /all/ subsets A of X.

It should be clear that any element of f(X\A) is in Y. To prove that any element of f(X\A) is not in f(A), one might try to derive a contradiction...
So if any element of f(X\A) were also in f(A), then that element must have two originals, one in A and one in X\A, and hence the function is not injective?

- Kamataat
 
  • #4
Yep, that's it.
 

1. What is the injective function property?

The injective function property, also known as the one-to-one property, is a property of a function in mathematics where each element in the domain maps to a unique element in the codomain. This means that no two elements in the domain can map to the same element in the codomain.

2. How can you prove that a function is injective?

To prove that a function is injective, you can use the "horizontal line test". This involves drawing horizontal lines across the graph of the function and checking if it intersects the graph at more than one point. If it does, then the function is not injective. Another method is to use algebraic proofs, such as showing that if f(a) = f(b), then a = b.

3. What are the benefits of having an injective function?

Having an injective function allows for a one-to-one correspondence between the elements in the domain and the codomain. This can be useful in various applications, such as creating unique identifiers or encoding information without loss of data.

4. Can a function be both injective and surjective?

Yes, a function can be both injective and surjective. This is known as a bijective function, where each element in the domain maps to a unique element in the codomain, and every element in the codomain has a corresponding element in the domain.

5. How is the injective function property used in real-world applications?

The injective function property is used in various real-world applications, such as in computer science for data encryption and compression, in data analysis for identifying unique data points, and in engineering for creating one-to-one mappings between input and output variables.

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