TITLE: Can we apply a function to a set? What does $f(A)$ mean when $A$ is a set of numbers?
QUESTION [8 upvotes]: For eg., let  $ A = \{-2, -1, 0, 1, 2 ,3\} $
Let $f(x) = \lfloor \frac {x^2}{3}\rfloor $
What is $ f(A) $ ?
In case anyone is wondering, this is homework, but I am not sure how to proceed as I can't find anything online regarding this.

REPLY [9 votes]: Yes, we can, but it's an abuse of notation. To be really strict (though only a set theorist or logician is likely to be this strict, hence why Peter answered in the negative!), you'd write $f"A$ or $f^{\to}(A)$, but it's clear from context what $f(A)$ is supposed to mean: the image of everything in $A$ under $f$.
In your case, you want $$f"A = \left\{\left\lfloor \frac{x^2}{3} \right\rfloor: x \in A\right\}$$
which is $\{0,1,3\}$.