Is it necessary to include "for all" when using set builder notation to define the image set?

Question

Image Set

Let F be a function from A to B, that is F is a subset from AxB. And let S be a subset of A.

Then the image set of S under F is: 0️⃣ F[S] := {y in B: there exist x in S such that xRy}

Is the above definition equivalent to:

1️⃣ F[S] := {all y in B such that there exist x in S such that xRy}

I claim definition 0️⃣ is not equivalent to 1️⃣ to show that let S={7,9} and B={2,4,8} and F={(7,2),(9,8)}

Then be 0️⃣ F[S] could be {2} or {8} or {2,8} but 1️⃣ must be {2,8}. This is because the ambiguity for “y in B” it could be every y in B in this case it is equivalent to 1️⃣ or some y in B which is not equivalent to 0️⃣ the core idea is ”y in B” could mean any y or every y in B.

Or we can say F[S]={y: there exist x in A such that xFy} without saying “y in B” or equivalently we can say F[S]=ran(F|S) where F|S means the restriction of F to S.

My whole idea is that “y in B” is ambiguous, and I don’t like that. Am I right?

Answer 1

Based on common conventions for set builder notation, I believe the "for all" or "any" is implied.

For example, to refer to the set $A$ of all real numbers from 0 to 5, it is common to write something like:

$$ A = \{ x : x \in \mathbb{R} \text{ and } 0 \leq x \leq 5 \}, $$

where it's clear that the set is meant to refer to the entire collection of number $[0,5]$, and not just one.

Personally, if I felt there was some risk of misunderstanding, due to context (e.g. conflating the image of a single element Vs. image of a set), I would include a "for all" or "any"; however, it is usually clear from context.

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[edit]

I got curious enough to check Wikipedia, according to their definition, the notation always implies inclusion of all elements that meet the criteria (emphasis added):

Set-builder notation can be used to describe sets that are defined by a predicate, rather than explicitly enumerated. In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a logical predicate. Thus there is a variable on the left of the separator, and a rule on the right of it. These three parts are contained in curly brackets: $$\{\ x \, | \, \Phi(x) \ \} \quad \text{ or }\quad \{ \ x : \Phi(x) \ \}$$ The vertical bar (or colon) is a separator that can be read as "such that", "for which", or "with the property that". The formula $\Phi(x)$ is said to be the rule or the predicate. All values of $x$ for which the predicate holds (is true) belong to the set being defined.

Of course, this is a convention, so it may vary from one source to another. FWIW I believe this is the only convention I've encountered.

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Edit 2: to clarify that the convention holds on both sides of the statement.

In the comments, you basically asked if the "for all" applies to both the variable and the logical predicate statement, and I would say yes.

Again, pointing to Wikipedia's entry regarding more complex expressions, an expression such as:

$\{ 2n \mid n \in \mathbb{N} \}$ where $\mathbb{N}$ is the set of all natural numbers, is the set of all even natural numbers.

Clearly here the assumption is that the set is the collection of all 2n.

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While I completely empathize with your confusion, (mathematical notation can feel very unintuitive when first encountered) I think if you stake a step back, you'll see that this is the obvious convention to adopt: if there is only one element in the set, there's no need for all this elaborate notation. The whole point of set builder notation is to efficiently and unambiguously refer to the members of a set that would be impractical (or impossible) to list, explicitly. If there were only one member, it would be much simpler (& unambiguous) to refer to that single member.

One of my best math teachers once told me that mathematical notation is founded on two principles:

• mathematicians are pedantic (i.e. they want to be precise and unambiguous), and - the element that makes it tricky for new-comers -
• they are lazy (i.e. they want to write the absolute minimum necessary).

This is a caricaturization, no doubt, but it can be very helpful, in situations like this :)