Problem Detail: Here is a definition from the functions section in my discrete math textbook (Discrete Mathematics and its Applications 7e, Rosen 2012):
Let $f$ be a function from $A$ to $B$, and let $S$ be a subset of $A$. The image of S under the function $f$ is the subset of $B$ that consists of the images of the elements of $S$. We denote the image of $S$ by $f(S)$, so: $f(S) = {t mid exists s in S (t = f(s)) }.$ We also use the shorthand ${ f(s) mid s in S }$ to denote this set.
So I guess I am just a little confused with the idea of an image. Is it the same as the codomain? And where it says, “The image of $S$ under the function …”. What do you think it means by “under”? Some examples would be great.
Asked By : Greg
Answered By : Nicholas Mancuso
The image is the subset of the co-domain that the domain actually maps to. Take for instance the following function $$f: mathbb{N} rightarrow mathbb{N}, quad f(x) = 5.$$ It is simply the constant function that always “returns” 5, no matter what natural number is “passed in.” The domain/co-domain defining $f$ is fine, but observe that the only value in the co-domain ever “used” is 5. In this case, the image of $f$ would just be the set ${5}.$ “The image of $S$ under $f$” is an explicit reference to the image of $f$ given $S$. That is to say, it is whatever each element in $S$ maps to as defined by $f$.
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