an image of a function is an element of a codomain or a subset of a codomain?
E.g. :ℕ →P(ℕ)−{∅} defined by ()={:∈ℕ and ≤≤2}
f(2) = {2,3,4}
or
f(2) = {{2,3,4}}
I am confused by the curly brackets, ()={ :∈ℕ and ≤≤2 }.
I am confused every time when a function is declared as f(y) = {...}, then my image must be a set?
A function $f$ from $A$ to $B$ is a "rule" which assigns to each $a\in A$ a unique $b\in B$. The element $b$ is written as $f(a)$. The set $A$ is called the domain and the set $B$ is called the codomain.
The image of $f$ is defined as $\operatorname{im}(f)=\{y\in B:\text{$y=f(x)$ for some $x\in A$}\}$. Since every element of the image of $f$ is an element of the codomain, this means that the image of $f$ is a subset of the codomain.
In your example, $f$ assigns to each natural number $x$, the set of natural numbers which are greater than or equal to $x$, and less than or equal to $2x$. Thus, for instance, $f(2)=\{2,3,4\}$. By contrast, the set $\{\{2,3,4\}\}$ is not a set of natural numbers: rather, it is a set containing a set of natural numbers. The set $\{2,3,4\}$ is an element of the image of $f$. The image of $f$ looks like this: $$ \operatorname{im}(f)=\{f(0),f(1),f(2),f(3),\dots\}=\{\{0\},\{1,2\},\{2,3,4\},\{3,4,5,6\},\dots\} \, . $$ (This is assuming that you consider $0$ to be a natural number; conventions vary.)
