MHB How do you find the image of the function?

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The image of the function f: Z*N -> R, defined as f(a,b) = a/b, is the set of rational numbers Q. This is because any rational number can be expressed as a ratio of an integer (a) and a positive integer (b). The function maps each pair of integers (m, n) to a rational number m/n, confirming that every rational number is included in the image. Additionally, the function is not one-to-one, as different pairs can yield the same rational number, but this does not affect the overall image. Thus, the image of the function is precisely the set of rational numbers.
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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.)
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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