How do you find the image of the function?

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SUMMARY

The function f : Z*N -> R defined by f(a,b) = a/b has an image that is precisely the set of rational numbers Q. This conclusion arises from the fact that any rational number can be expressed as the ratio of an integer (a) and a positive integer (b). The image of a set A under the function f is defined as { y | y = f(x), x ∈ A }, confirming that every rational number corresponds to a unique pair (m, n) where m is an integer and n is a positive integer. The function is not one-to-one, as demonstrated by the example f(2, 4) = f(1, 2), but this characteristic does not affect the determination of the image.

PREREQUISITES
  • Understanding of functions and their images in mathematical terms
  • Familiarity with rational numbers and their properties
  • Knowledge of integer and positive integer definitions
  • Basic grasp of set theory and function notation
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  • Study the properties of rational numbers and their representation as fractions
  • Explore the concept of function images in more depth
  • Learn about one-to-one and onto functions in mathematical analysis
  • Investigate the implications of integer and positive integer domains in function mapping
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Mathematicians, educators, and students studying functions, rational numbers, and set theory will benefit from this discussion.

WannaBe
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What is the image of the function f : Z*N -> R ; f(a,b)= a/b?

I know the answer is Q (rational numbers) but I don't know how to find it.
 
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WannaBe said:
What is the image of the function f : Z*N -> R ; f(a,b)= a/b?
I know the answer is Q (rational numbers) but I don't know how to find it.

You surely know that any rational number is the ratio of two integers.
Note that any rational can be written as an integer divided by a positive integer.
Here let $$\mathbb{N}=\mathbb{Z}^+$$.
 
Technically, you don't "find the image" of a function, you find the image of a set by a function: the image of set A, by function f, is \{ y| y= f(x), x\in A\}.

As Plato said, the set of rational numbers is defined as the set of all fractions, \frac{a}{b} with a any integer, b any positive integer (taking the denominator from the positive integers let's us assign the sign of the fraction to the numerator and voids division by 0).

Now, if y is any rational number, there exist integer, m, and positive integer, n, such that y= m/n. That is precisely f(m, n) so every rational number is in the image.

Conversely, for any pair, (m, n), m an integer, n a positive integer, f(m,n)= m/n is a rational number so the image is precisely the set of rational numbers.

(This function is NOT "one-to-one", f(2, 4)= f(1, 2), but that is not relevant to this problem.)
 

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