Inverse functions and null set.

In summary, the conversation discusses the concept of an inverse function and whether one exists in a given scenario. The speaker also raises questions about the requirements for a function to have an inverse and the possibility of using a null set in the domain. The conversation concludes that while the given function may not be a function from A to B, it is still possible to define an inverse function from a subset of A to B.
  • #1
blastoise
22
0
Ok,

I understand an inverse function sends a variable in the range to the corresponding value in the domain, but am not sure if what I'm thinking is correct... : For example:

Let A be the set


[itex] A = \{1,2,3,7,8\} ; B = \{4,5,6\} [/itex] and the function [itex] f [/itex] map A to B s.t

f(1) = 4
f(2) = 5
f(3) = 6

so 7,8 do not have a value that is mapped one to one.

I understand f is an surjection, but not a injection . But, does an inverse function exist?

I would say yes, despite there not being a value in B that maps to 7 or 8.
Is my thinking correct?


Also, am I correct to say that a function does not have to use every element in the domain in order to have an inverse; I am confused because wouldn't one just say it maps to the null set and the inverse of the null set would contain values(7,8) that it maps to and hence not a function...?
 
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  • #2
blastoise said:
Ok,

I understand an inverse function sends a variable in the range to the corresponding value in the domain, but am not sure if what I'm thinking is correct... : For example:

Let A be the set


[itex] A = \{1,2,3,7,8\} ; B = \{4,5,6\} [/itex] and the function [itex] f [/itex] map A to B s.t

f(1) = 4
f(2) = 5
f(3) = 6

so 7,8 do not have a value that is mapped one to one.


*** ...and thus f is NOT a function from A to B but one from a proper subset of A to B...***



I understand f is an surjection, but not a injection . But, does an inverse function exist?


*** Yes, if we look at f defined from {1,2,3} onto {4,5,6}. ***



I would say yes, despite there not being a value in B that maps to 7 or 8.
Is my thinking correct?


Also, am I correct to say that a function does not have to use every element in the domain in order to have an inverse; I am confused because wouldn't one just say it maps to the null set and the inverse of the null set would contain values(7,8) that it maps to and hence not a function...?


If we talk of a function [itex]f: A\to B[/itex] , we're explicitly assuming f is defined in the whole of A. Whether f is 1-1 and/or onto is another matter.

DonAntonio
 

FAQ: Inverse functions and null set.

1) What are inverse functions and how are they defined?

Inverse functions are functions that "undo" each other. This means that if we have a function f(x), its inverse function will be able to return us to the original input x. In other words, if we have f(x) = y, then the inverse function, denoted as f-1(y), will give us back x.

2) How can we find the inverse of a function?

To find the inverse of a function, we can use the following steps:

  1. Replace f(x) with y.
  2. Swap the x and y variables.
  3. Solve for y.
  4. Replace y with f-1(x) to get the inverse function.

3) Can every function have an inverse?

No, not every function has an inverse. A function must be one-to-one, meaning that each input has a unique output, in order to have an inverse. If there are multiple inputs that give the same output, then the inverse would not be able to "undo" the function.

4) What is the null set and how is it related to inverse functions?

The null set, also known as the empty set, is a set that contains no elements. Inverse functions have a null set when the function does not have an inverse. This means that there are no inputs that can return a specific output, making the inverse function undefined.

5) How can we determine if a function has an inverse by looking at its graph?

A function will have an inverse if it passes the horizontal line test. This means that if we draw a horizontal line through the graph, it should only intersect the function at most once. If the line intersects the function more than once, then the function does not have an inverse.

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