Set theory, functions, bijectivity

In summary, Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects or elements. Functions, a mathematical concept that relates inputs to outputs, can be bijective, meaning they have a one-to-one and onto mapping between their domain and range. This is determined using the horizontal line test and is important in mathematics because it guarantees a one-to-one correspondence between two sets and allows for the use of inverse functions.
  • #1
Daveyboy
58
0
f:X[tex]\rightarrow[/tex]Y, A[tex]\subset[/tex]X
f(Ac)=[f(A)]c implies f bijective.

Just trying to apply the definitions of injective and bijective. The equivalence makes sense but I am having a hard time showing it.

f(x)=f(y) implies x=y and for every y in Y there exists a x in X s.t. f(x)=y.

I mean all I have is if f(x)=f(y)
then f(X\x)=Y\f(x\x)=f(X\y)-Y-f(X\y)...
 
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  • #2
what's the question?
 
  • #3
Daveyboy said:
f(Ac)=[f(A)]c implies f bijective.
this implication
 

What is set theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects or elements. It is used to define and analyze mathematical concepts such as functions, numbers, and algebraic structures.

What is a function?

A function is a mathematical concept that relates inputs (called the domain) to outputs (called the range) in a consistent and predictable way. Each input has only one corresponding output, and every output has at least one corresponding input.

What does it mean for a function to be bijective?

A function is bijective if it is both injective and surjective. This means that every element in the range has a unique corresponding element in the domain, and every element in the domain has a corresponding element in the range. In other words, a bijective function has a one-to-one and onto mapping between its domain and range.

How do you determine if a function is bijective?

To determine if a function is bijective, you can use the horizontal line test. If a horizontal line intersects the graph of the function at exactly one point, then the function is injective. If the horizontal line intersects the graph at every point, then the function is surjective. If both conditions are met, then the function is bijective.

Why is bijectivity important in mathematics?

Bijectivity is important in mathematics because it guarantees a one-to-one correspondence between two sets. This means that information can be easily and accurately transferred between the two sets without any loss or duplication. It also allows for the use of inverse functions, which are crucial in solving many mathematical problems.

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