Undergrad Question regarding quantifier statement

[CGandC]

TL;DR

.

Suppose I have the following ( arbitrary ) statement:
$$ \forall x\in{S} \ ( P(x) ) $$
Which means: For all x that belongs to S such that P(x).

Can I write it as the following so that they are equivalent? ( although it is not conventional ):
$$ \forall x\in{S} \land ( P(x) ) $$
Can I write this way? what does this mean? ( in plain english it is ambiguous : All x that belongs to S and also P(x) )My reasoning is as follows:
$$ \forall x\in{S} \\ $$ is a statement so I denote it as Q ( not Q(x) since x is a bound variable in this statement ) .
So if i'll look back at the initial statement, i'll write it as follows: $$ Q , ( P(x) ) $$ which I write as $$ Q \land ( P(x) ) $$

Thanks for any help in advance.

 

[SSequence]

Suppose I have the following ( arbitrary ) statement:
$$ \forall x\in{S} \ ( P(x) ) $$
Which means: For all x that belongs to S such that P(x).

Can I write it as the following so that they are equivalent? ( although it is not conventional ):
$$ \forall x\in{S} \land ( P(x) ) $$
Can I write this way? what does this mean? ( in plain english it is ambiguous : All x that belongs to S and also P(x) )

You could write the following statement as equivalent to first one:
$\forall x \,\, (x \in {S} \rightarrow P(x) )$
This does assume a bit of context though (discussed below).

Generally speaking whenever something like $\forall x$ is written instead of, for example: $\forall x \in \mathbb{R}$, $\forall x \in \mathbb{Q}$, $\forall x \in \mathbb{N}$, $\forall x \in \mathrm{Ord}$ etc. ... that means the domain over which the quantification is taking place in $\forall x$ is (presumably) understood from the context.

==============

As an example, if we were considering real numbers and we had $S \subseteq R$, then we could write either of the following:
$\forall x\in{S} \ ( P(x) )$ ------1
$\forall x \in \mathbb{R} \,\, (x \in {S} \rightarrow P(x) )$ ------2

However, if it was reasonably clear from the context that our domain of discourse is clearly real numbers only [or actually, any superset of real numbers ... or really, even any superset of $S$], then we could write it as above:
$\forall x \,\, (x \in {S} \rightarrow P(x) )$ ------3

The point being that sentences-(1),(2) are logically equivalent (well in classical logic on the very least). If the domain of quantification is understood to be some super-set of $\mathbb{R}$, then sentence-(3) is also equivalent to both (1),(2).

Edit:
To elaborate a bit with regards to the statement you wrote:
$\forall x\in{S} \land ( P(x) )$
I am not certain about validity of writing this since, I think, when we write something like $\forall x\in{S}$, it is expected to be followed by a predicate. It seems to me that normally this will not be considered valid, or at least, a clear way to write.

Though this kind of expression seems useful in defining sets if you remove the initial $\forall$ and write the following open sentence (in one variable):
$x\in{S} \land ( P(x) )$
[See end of post#4 for example]

Edit2:
Sorry I mixed up the "and" and "implication" in the initial version of the post. I think it should probably be OK now (I will re-check). Re-wrote a few further points.

 

[Stephen Tashi]

Suppose I have the following ( arbitrary ) statement:
$$ \forall x\in{S} \ ( P(x) ) $$
Which means: For all x that belongs to S such that P(x).

That is a sentence fragment, not what grammarians call a "complete sentence". So it doesn't describe a statement in the sense of a proposition. If it were a proposition then it could be interpreted as being either True or False.

Perhaps your question concerns different ways in which a set can be defined. Try to phrase it that way.

Can I write it as the following so that they are equivalent? ( although it is not conventional ):
$$ \forall x\in{S} \land ( P(x) ) $$
Can I write this way?

If $S$ is a given set, you can define the set $V$ by saying
$\forall x \{ x \in V$ if and only if $((x\in S) \land P(x)) \}$.

There are different schemes of mathematical notation. Some people may use the notation "$\forall x \in S$" because speaking the names of the individual symbols produces a phrase that can be interpreted in English. However, I haven't seen any formal system of notation where "$\forall x \in S \land P(x)$" is correct syntax. Usual styles would be "$(\forall x) \{ x \in S \land P(x)\}$" or "$(\forall x \in S)\{ P(x)\}$".

what does this mean? ( in plain english it is ambiguous : All x that belongs to S and also P(x) )

What do you mean by "it" being ambiguous? Do you mean that the phrase "All x that belongs to S and also P(x) " can be interpreted to define two distinct sets? What are those two distinct sets?

 

[SSequence]

That is a sentence fragment, not what grammarians call a "complete sentence". So it doesn't describe a statement in the sense of a proposition. If it were a proposition then it could be interpreted as being either True or False.

I don't quite understand this. The expression $\forall x\in{S} \ ( P(x) )$ is a "closed sentence" (so to speak). In classical sense, it would have a true/false value.

It is true that this is informal way of writing (but it seems that this is common).

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One other point should be mentioned, I think. When we write an expression like:
$(\forall x) \{ x \in S \land P(x)\}$
The problem is that this is a false sentence if our "domain of discourse" [i.e. the "world" over which quantification is taking place] is anything that is a strict super-set of $S$ [or e.g. even in set theory where the quantification is taking over a class].

If $S$ is a given set, you can define the set $V$ by saying
$\forall x \{ x \in V$ if and only if $((x\in S) \land P(x)) \}$.

That's right. One other way I have seen a definition of $V$ is something like:
$V=\{x \in S\,|\, P(x) \}$

And if the domain of discourse was implicitly understood (say over class of sets etc.) then we can also write:
$V=\{x\,|\, x \in S \, \land \, P(x) \}$

 

[SSequence]

Regarding the formalization of sentence like:
$\forall x\in{S} \ ( P(x) )$
Here is a sketch of how this should work out probably (at least that's my guess/understanding of it). I hope this won't confuse the OP (as it is an extra detail and not directly related to question, which I tried to answer in post#2).

So, as far as the question in OP is concerned, this post can be ignored.

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Note that below we are strictly assuming the language of set theory. First we re-write $\forall x\in{S} \ ( P(x) )$ as (informal expression):
$\forall x (x\in{S} \rightarrow \ P(x) )$

Now we would have a (valid) open predicate $Q(x)$ (in one variable) such that for any arbitrary set $a$, we have $a \in S$ iff $Q(a)$ is true. We re-write the above expression as (informal expression):
$\forall x (Q(x) \rightarrow \ P(x) )$
When we substitute the formulas of the predicates $Q$ and $P$ in previous expression we would get the formal expression (in set theory language) which corresponds exactly to $\forall x\in{S} \ ( P(x) )$.

 

[Stephen Tashi]

I don't quite understand this. The expression $\forall x\in{S} \ ( P(x) )$ is a "closed sentence" (so to speak). In classical sense, it would have a true/false value.

I agree that "For each x in the set S, x has property P" is a "closed sentence".

However the OP's interpretation of his notation was:

For all x that belongs to S such that P(x)

which isn't a phrase that I can assign a truth value.

 

[SSequence]

Your wording is correct but the OP's wording (which you quoted) also seems reasonable to me. I have seen similar wording being used.

For example, I have seen both wordings being used:
"For all x that belongs to S such that P(x)"
"For all x that belongs to S such that P(x) is true"

 

[Stephen Tashi]

For example, I have seen both wordings being used:
"For all x that belongs to S such that P(x)"
"For all x that belongs to S such that P(x) is true"

I've seen such phases used as part of a complete sentence. My point is that, by themselves, they don't assert anything that can be judged true or false.

For example the phrase "For all x in the set of U.S. Citizens such that x is 18 years old" does not convey the idea that "Each U.S. Citizen is 18 years old".

 

[CGandC]

Thanks for the helpful comments, I understand and agree with everything said above.

What do you mean by "it" being ambiguous? Do you mean that the phrase "All x that belongs to S and also P(x) " can be interpreted to define two distinct sets? What are those two distinct sets?

Meaning that it on the one hand sounds like " x belongs to S and also belongs P(x) " and on the other hand it means that " x belongs only to set S [ and does not belong to P(x) ] , but P(x) is also satisfied ".

Actually I think writing $\forall x\in{S} ( P(x) )$ as $\forall x\in{S} \land ( P(x) )$ is a mistake ( or does not make sense ) since this would entail that $\forall x\in{S}$ stands for some statement $Q$ . And $P(x)$ stands for some other statement.
So that $\forall x\in{S} \land ( P(x) )$ can be written as $Q \land P(x)$.
But from looking at the last syntax, we infer that x is not bound variable ( also called dummy variable ) but a free variable ( since $P(x)$ depends on x ).
But since in the original statement ( $\forall x\in{S} ( P(x) )$ ) , x was a bound variable, therefore we reach a contradiction.
Hence, writing $\forall x\in{S} ( P(x) )$ as $\forall x\in{S} \land ( P(x) )$ is false.

Edit:
To elaborate a bit with regards to the statement you wrote:
$\forall x\in{S} \land ( P(x) )$
I am not certain about validity of writing this since, I think, when we write something like $\forall x\in{S}$, it is expected to be followed by a predicate. It seems to me that normally this will not be considered valid, or at least, a clear way to write.

That's why I asked the question, I didn't find anything on the validity of writing $\forall x\in{S} \land ( P(x) )$ ( nor does it make much sense if I really try to interpret it ). Well now I conclude that I should avoid writing the way I asked about.

Normally I would write $\forall x\in{S} ( P(x) )$ as $\forall x \,\, (x \in {S} \rightarrow P(x) )$

 

[SSequence]

That's why I asked the question, I didn't find anything on the validity of writing $\forall x\in{S} \land ( P(x) )$ ( nor does it make much sense if I really try to interpret it ). Well now I conclude that I should avoid writing the way I asked about.

Normally I would write $\forall x\in{S} ( P(x) )$ as $\forall x \,\, (x \in {S} \rightarrow P(x) )$

Yeah, $\forall x \,\, (x \in {S} \rightarrow P(x) )$ should be fine.

It seems to me that it might be better to avoid notation like $\forall x\in{S} \land ( P(x) )$ anyway.

If we try to write (by trying to "interpret" it anyway):
$\forall x (x\in{S}) \land ( P(y) )$
it still doesn't seem to be of much use. Since now we have an open sentence which is false for all possible values of $y$ (assuming domain of discourse to be strict super-set of $S$).

 

[Stephen Tashi]

and also belongs P(x) "

It's important to distinguish between sets and statements. The usual interpretation of "$P(x)$" is "$x$ has property $P$". So "$P(x)$" denotes a propositional function. It does not denote a set.

We can define a set $V$ by saying "$V$ is the set of things which have property $P$". Strictly speaking, $P(x)$ is not a notation for the set $V$.

A statement that a variable has a property is not the same as the set of things that have the property.