Ambiguity in the notation $f(A)$

Question

Let $f:X\to Y$ be a map. When $A\in X$, we write the image of $A$ under $f$ as $f(A)$. On the other hand, when $A\subseteq X$, we write $\{f(a)\mid a\in A\}$ also as $f(A)$.

But this obviously creates some ambiguity: when $A$ is both an element and a subset of $X$, what does $f(A)$ mean then?

For example, let $X=\{1,\{1\}\}$, $Y=\{a,b\}$ and $f$ be defined by $f$ maps the element $1$ of $X$ to $a$ and the element $\{1\}$ to $b$. Then does $f(\{1\})$ mean $\{a\}$ or $b$?

I do not understand why such ambiguity still exists in elementary set theory.

Answer 1

It is ambiguous and can have either meaning, depending on context. Most of the time the intended meaning is clear, since unless you're doing axiomatic set theory you're not likely to encounter a set $A$ which you want to think of as both an element of and a subset of another set $X$. An alternate notation that is often used by set theorists (though not as much by general mathematicians) to avoid this ambiguity is to write $f[A]$ rather than $f(A)$ for $\{f(a):a\in A\}$.

Answer 2

Yes, the notation $f(A)$ could, potentially, be ambiguous, since we use $f(A)$ both for the value of a function at a point $A$, as well as for the image of a set $A$. However, I don't agree with the premise of the question—namely, that this potential ambiguity is something which needs to be taken terribly seriously. Indeed, I would argue that the situation described in the question is incredibly contrived, and unlikely to occur or cause any problems in actual mathematical writing or reasoning.

That begin said, if you are working in a setting where this ambiguity is a real problem, then it is important to recall that all notation in mathematics is arbitrary, and is agreed to by convention. If standard notation is causing problems, then it is entirely reasonable to introduce new notation. For example:

Definition: Let $f : X \to Y$ (so that $f \subseteq X \times Y$ and $(x,y_1), (x,y_2) \in f$ implies that $y_1 = y_2$).

• For $x \in X$, denote by $f(x)$ the value of $f$ at $x$. That is, if $(x,y) \in f \subseteq X\times Y$, then write $f(x) = y$.

• For $A \subseteq X$, denote by $f[x]$ the image of $A$. That is $$ f[A] := \{ f(a) : a \in A \}, $$ where $f(a)$ denotes the value of $f$ at $a$, as above.

For the record, the notation suggested here is, actually, relatively common. For example, it is described on Wikipedia. Other possible notations which would likely be well understood (with an appropriate definition or explanation):

• For the value of a function $f$ at a point $A$ in the domain of $f$, we might write $f(x)|_{x = A}$. If this notation were adopted, then $f(A)$ could be retained to exclusively denote the image of a set $A \subseteq X$.
• For the image of a subset of the domain of $f$, we might write (for example) $$ \operatorname{Range}(f|_A),\qquad \mathscr{R}(f|A),\qquad \operatorname{Image}(A),\qquad\text{or}\qquad \operatorname{Im}_f(A). $$ The first two notations regard the image of $A$ under $f$ as the range of the restriction of $f$ to $A$ (either $f|_{A}$ or $f|A$ might denote this restriction). The third notation calls a spade a spade, but suppresses the dependence on $f$. The last notation abbreviates the previous, but recovers the dependence on $f$. If any of these notations were to be adopted, the notation $f(A)$ could be retained for the value of $f$ at a point $A \in X$.

Answer 3

Let $f:X\to Y$ be a map. When $A\in X$, we write the image of $A$ under $f$ as $f(A)$. On the other hand, when $A\subseteq X$, we write $\{f(a)\mid a\in A\}$ also as $f(A)$.

The image of $A$ under $f$ is $\{f(a)\mid a\in A\}$. For $A\in X$, you seem to be intending to be talking about $f$ evaluated at $A$. This is the primary meaning of $f(A)$. The $\{f(a)\mid a\in A\}$ meaning is simply shorthand, and if there is any conflict between the two meanings, the first predominates.