I needed some help in understanding how to determine the image of a function with a Cartesian product domain.
For example a question like this:
$$f: \mathbb{N}^+ \times \mathbb{N}^+ \longrightarrow \mathbb{R}, \quad f(a, b) = (a+b)/2 .$$
How would I find the image of this function, or, if given the image, prove that it is the image?
My confusion is due to the two different variables $a$ and $b.$ Any help would be appreciated. Thank you.
One way, which is often how I approach such problems, is to consider what you get if you fix $a$ and then let $b$ vary, to get a partial image $I_a$. Then think about repeating for a different $a$, and so on. Then try to think about what the union $$\bigcup_{a}I_a$$ of all the partial images is.
Another approach is to realize that $$\{a+b\mid a,b \in \mathbb N\} = \{2,3,4,5,6,\ldots\}$$
Can you then see what halving each element then gives?