##A \subset B \iff \exists x \in B \land x \not\in A##?

[yucheng]

TL;DR Summary

Is $A \subset B \iff \exists x \in B \land x \not\in A$ correct?
N.B. I ommitted the other half of the statement for emphasis, see the complete statement below.

$A \subset B$ means that $\exists x \in B$ such that $x \not\in A$. Is this logically equivalent to $\exists x \in B \land x \not\in A$? Formally, $$A \subset B \iff {(\exists x \in B \land x \not\in A)\land(\forall y \in A \land y\in B)}$$

I have tried consulting https://math.stackexchange.com/questions/432345/such-that-logical-symbol, it seemes that there is no logical notation for "such that". I would thus like to confirm whether the above statement is correct (I mean whether it fulfils the notational convention).

Also, is this statement true and equivalent to the statement above: $$A \subset B \iff {(\exists x \in B \land x \not\in A)\land(\forall x \in A \land x\in B)}$$ I understand that it is bound to cause confusion, since I used "x" in two different, concurrent statements, but is the "x" only circumscribed to each of the simple statements that form the compound statement, or is it somewhat like a global variable, such that I should use a different variable (in this case y) for different statements? Thanks in advance!

 

[member 587159]

Disclaimer: I assume $A \subset B$ means strict inclusion. I would write this as $\subsetneq$.

You have to make clearer on what the quantifier works (use brackets).

But in any case, what you wrote cannot be correct because nothing in your statement garantuees that you have an inclusion. You should rather have something like

$A \subset B \iff (\forall x : (x \in A \implies x \in B)) \land (\exists y \in A: y \notin B)$

 

[PeroK]

@yucheng $A \subset B$ means that all the elements of $A$ are in $B$. Your statements appear to have nothing to do with that.

 

[yucheng]

Disclaimer: I assume $A \subset B$ means strict inclusion. I would write this as $\subsetneq$.

I for got to clarify this. Sorry!

You have to make clearer on what the quantifier works (use brackets).

@Math_QED What do you mean by on what the quantifier works by use of brackets? At least I see where you're coming from after checking with wikipedia:

∃ x: P(x)

I guess this is what you mean, using the colon? Never mind, I get what you mean if the bracket you are referring to is for the statement $\forall x:(x\in A...$.

$A \subset B \iff (\forall x : (x \in A \implies x \in B)) \land (\exists y \in A: y \notin B)$

Umm, should the statement after and be $\exists y \in B: y \notin A$, otherwise $A\not\subseteq B$?

Lastly, does ":" mean such that?

@PeroK That was what I thought too!

 

[PeroK]

What about: $$A \not \subset B \iff \exists x : (x \in A \land x \notin B)$$

 

[member 587159]

What about: $$A \not \subset B \iff \exists x : (x \in A \land x \notin B)$$

$A \subsetneq B$ and $A \not\subseteq B$ are two different things.

 

[PeroK]

$A \subsetneq B$ and $A \not\subseteq B$ are two different things.

Okay, but when you write \subset in Latex, you get $\subset$. AFAIK that's the convention Latex uses. If I wasn't using Latex, it would be different.

 

[member 587159]

I for got to clarify this. Sorry!


@Math_QED What do you mean by on what the quantifier works by use of brackets? At least I see where you're coming from after checking with wikipedia:

∃ x: P(x)

I guess this is what you mean, using the colon? Never mind, I get what you mean if the bracket you are referring to is for the statement $\forall x:(x\in A...$.


Umm, should the statement after and be $\exists y \in B: y \notin A$, otherwise $A\not\subseteq B$?

Lastly, does ":" mean such that?

@PeroK That was what I thought too!

What I mean with bracketing: You write $\exists x \in B \land x \notin A$. This is not entirely correct. Here are some options:

(1) $(\exists x \in B) \land (\exists x \not\in A)$ -> $B$ and $A$ are non-empty;
(2) $\exists x: (x \in A \land x \notin B)$ -> There is an element $x \in A$ that is not in $B$. You can also write this last condition in short as $\exists x \in A \setminus B$ or simply $A \setminus B\neq \emptyset$, but I guess you want to write everything in quantifiers.

And yes, the ":" sign can be read as "such that". You don't need to write it though. For example,

$\forall x : P(x)$

and

$\forall x P(x)$

mean the same thing, but I think the former is more pleasant to read.

 

[PeroK]

It just seems to me that the original post was much closer to a statement that A is not a subset of B than A is a subset of B. Maybe that's the source of the confusion?

 

[yucheng]

It just seems to me that the original post was much closer to a statement that A is not a subset of B than A is a subset of B. Maybe that's the source of the confusion?

Actually, I was just trying to express "all elements in A are elements of B", "and" " there is some element of B that is not in A", i.e. $A\subsetneq B$. @Math_QED was correct.

 

[PeroK]

Actually, I was just trying to express "all elements in A are elements of B", "and" " there is some element of B that is not in A", i.e. $A\subsetneq B$. @Math_QED was correct.

Okay, but you fooled me!

 

[SSequence]

Disclaimer: I assume $A \subset B$ means strict inclusion. I would write this as $\subsetneq$.

I think the former symbol is pretty standard (sometimes read as "proper subset"). The latter one is also fairly self-explanatory though. I think both symbols are fine.

$A \subset B \iff (\forall x : (x \in A \implies x \in B)) \land (\exists y \in A: y \notin B)$

I think that in the second part you might have got the order of A and B in reverse? Shouldn't it be like below (since we want there to be an element in B which is not in A)?
$A \subset B \iff (\forall x : (x \in A \implies x \in B)) \land (\exists y \in B: y \notin A)$

 

[member 587159]

I think the former symbol is pretty standard (sometimes read as "proper subset"). The latter one is also fairly self-explanatory though. I think both symbols are fine.

I think that in the second part you might have got the order of A and B in reverse? Shouldn't it be like below (since we want there to be an element in B which is not in A)?
$A \subset B \iff (\forall x : (x \in A \implies x \in B)) \land (\exists y \in B: y \notin A)$

I think the notation $\subset$ should be avoided because it is ambiguous. I know mathematicians that use it to mean $\subseteq$ while others use it to mean $\subsetneq$. Of course, this doesn't matter as long as the author specifies what notation he/she is using.

And of course you are right that I reversed the order. Thanks for noticing.

 

[SSequence]

I think the notation $\subset$ should be avoided because it is ambiguous. I know mathematicians that use it to mean $\subseteq$ while others use it to mean $\subsetneq$. Of course, this doesn't matter as long as the author specifies what notation he/she is using.

Personally I would only use $\subset$ to mean "proper subset". But I remember something interesting. I was reading an introductory text on topology (about eight months or ago). I could only go through the first few sections of first chapter in three weeks or so (it was a good text). I didn't get the time to read beyond that.

But anyway, coming to point, the text was using $\subset$ to mean $\subseteq$. I was interpreting $\subset$ simply as "proper subset" (since that's the only way I had ever seen the symbol being used). It was a bit strange to me that some of the definitions would seem "weird/unapplicable" for boundary cases. It took me a week or two to realize the author was using $\subset$ to mean just "subset" and not "proper subset".

So actually, it is a good advice that whenever the symbol $\subset$ is being used (e.g. in a book) one should be bit wary that the author might just mean "subset" and not "proper subset".

 

[PeroK]

Personally I would only use $\subset$ to mean "proper subset".

I would agree, but Latex renders \subset as $\subset$. Unless that is changed, ambiguity on this forum (or any forum using Latex) is unavoidable.

 

[sysprog]

I would agree, but Latex renders \subset as $\subset$. Unless that is changed, ambiguity on this forum (or any forum using Latex) is unavoidable.

You can use \subsetneq to unambiguously indicate a proper subset:
$A \subsetneq B$

Or you can use an expression like this, in which brackets are employed to establish the scope of the quantifiers:

$\forall x [(x \in A) \Rightarrow (x \in B)] \wedge \exists x [(x \in B) \wedge (x\not\in A)]$

For the existential quantifier, the 'such that' meaning applies to the expression that is inside the brackets.