Proving One-to-One Functions: An Introduction to Counterexamples

In summary, there is a discussion about two functions: f: Z --> Z defined as f(n) = |2n-1| and g: {n element of Z|n>=10} --> Z defined as g(n) = |2n-1|. The task is to prove that f is not one to one and g is one to one. It is suggested to use a graph to visualize the functions and to prove that a function is not one to one, a counterexample where f(x1)=f(x2) is true and x1=x2 is false must be provided. The conversation also mentions the need to have a basic understanding of cartesian products of sets and relations in order to discuss functions.
  • #1
cue928
130
0
I am being asked to prove the following:
f: Z --> Z by f(n) = |2n-1| is not one to one and g:{n element of Z|n>=10} --> Z with g(n) = |2n-1| is one to one. Can anyone help me get started on this? The example done in class involved substituting in and finding out if the values were equal. Maybe it's the absolute value bars, not sure how to prove this in this instance.

Incidentally, can you have a discussion about functions without having discussed cartesian products of sets and also relations?
 
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  • #2
A graph should help you see what's going on -- even if you can't use it in your formal solution.
 
  • #3
see when function is one to one it means that whenever you have f(x1)=f(x2) it implies x1=x2...

so to prove that a function is NOT one to one, you have to come up with a counterexample where f(x1)=f(x2) is true and x1=x2 is false.
now for function f , what can you tell about the ordered pairs (0,1) and (1,1) ?
 

FAQ: Proving One-to-One Functions: An Introduction to Counterexamples

What does it mean for a function to be one-to-one?

One-to-one, also known as injective, means that each input value has a unique output value. This means that no two different input values can produce the same output value. In other words, every element in the domain of the function is mapped to a unique element in the range.

How can you prove that a function is one-to-one?

To prove that a function is one-to-one, you can use the horizontal line test. This is done by drawing a horizontal line on the graph of the function. If the line intersects the graph at more than one point, then the function is not one-to-one. If the line only intersects the graph at one point, then the function is one-to-one.

Is every one-to-one function also onto?

No, not every one-to-one function is also onto. A function is onto, also known as surjective, if every element in the range of the function is mapped to by at least one element in the domain. This means that there are no "leftover" elements in the range that are not mapped to by any element in the domain. A one-to-one function can have elements in the range that are not mapped to by any element in the domain, thus making it not onto.

Can a function be both one-to-one and onto?

Yes, a function can be both one-to-one and onto. A function that is both one-to-one and onto is called a bijective function. This means that every element in the domain is mapped to a unique element in the range, and every element in the range is mapped to by at least one element in the domain.

How does the inverse function relate to one-to-one functions?

A function that is one-to-one has an inverse function. The inverse function is obtained by switching the input and output values of the original function. This means that the inverse function will map every output value of the original function back to its corresponding input value. In other words, the inverse function "undoes" the original function. However, not all functions have an inverse, and a function can only have an inverse if it is one-to-one.

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