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I'm not so bad in math however I want to understand math the way mathematicians do, so I picked up this book How to Think Like a Mathematician. The book seems the book that I'm looking for. Definitely, it gonna be very long dark tunnel to me until I see the light. Any ways, he started with sets and I came across the following

1- For any set $X$, we have $X \subseteq X$.

2- For any set $X$, we have $\emptyset \subseteq X$.

I couldn't imagine what do the two aforementioned statements imply. For the first one, how come a set is part of or equal itself? I can imagine the equality here but not the partition. The second statement, how come a non-empty set has an empty set? For example, let $ X = \{1, 2, 3\}$, now why $\emptyset \subseteq X$?

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If you already understand why $X\subseteq X$ (because the little underline means that it is possible that $X=X$), then the other question has been thoroughly chewed all across the site. –  Asaf Karagila Jun 7 at 11:06
    
Of course, some authors skip the underline and write $X \subset X$ to mean "X is a subset of (possibly equal to) X," and would write $\{1\} \subsetneq \{1, 2\}$ to mean that the first set is a proper subset of (not equal to) the second. –  Strants Jun 8 at 1:26
    
@AsafKaragila, no I'm not understanding $ X \subset X$ however $X = X$ makes sense. –  CroCo Jun 8 at 1:31
    
I didn't use $X\subset X$ on purpose there. –  Asaf Karagila Jun 8 at 6:42

5 Answers 5

up vote 17 down vote accepted

A very good principle when starting out in any new area of maths: get really, really clear about key definitions. Don't just make a stab at guessing (or go by the perhaps misleading everyday terminology used in labelling, or by an informal gloss on the definition): look carefully at how key notions are rigorously defined.

Now, what does $X \subseteq Y$ mean, officially?

$X \subseteq Y$ holds, by definition, if and only if, for any $x$, if $x \in X$ then $x \in Y$.

Informally, yes, you might be tempted to gloss $X \subseteq Y$ as e.g. "$X$ is a part of $Y$"; that can be useful, but as you are finding, it can also be misleading. So forget for a moment the rough informal gloss and concentrate on the official definition.

Applied to your two cases, we have first, by definition

$X \subseteq X$ if and only if, for any $x$, if $x \in X$ then $x \in X$.

Well now ask: Is it the case that for any $x$, if $x \in X$ then $x \in X$? And so what does that tell you about $X \subseteq X$?

Second, we have by definition

$\emptyset \subseteq X$ if and only if, for any $x$, if $x \in \emptyset$ then $x \in X$.

Well now ask: Is it the case that for any $x$, if $x \in \emptyset$ then $x \in X$? Recall that $x \in \emptyset$ is always false: what happens to quantified conditionals where the antecedent is always false? And so what does that tell you about $\emptyset \subseteq X$?

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I found it hard to understand the explanation of the second one. Can you explain it any more clearly? –  AaKASH Jun 7 at 13:15
    
@AaKASH: Recall that $\emptyset$ is empty (has no elements). So $x\in\emptyset$ is always false... there is nothing in $\emptyset$ so surely $x$ is not in there! Now any implication like "if A then B" in which A is false is always true. So if I say "If $x\in\emptyset$, then I have cats for ears," that implication is (vacuously) true because my hypothesis is false. –  5space Jun 7 at 16:10
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+1 for the first paragraph alone. –  Michael Albanese Jun 8 at 3:40

Both questions really boil down to how mathematicians use logic.

1.) $X \subseteq X$ means that for all elements of $X$ they are elements of $X$.

2.) $\emptyset \subseteq X$ means that for all elements of $\emptyset$ they are elements of $X$.

To see why 1.) is true, consider an element $x$ in the set $X$, well then we know that $x$ is in $X$ (!) and so we have shown that all elements from $X$ are also elements of $X$.

To see why 2.) is true, we must show that given an element of $\emptyset$ it must belong to $X$. But, there are no elements of $\emptyset$ and so the implication is (trivially) satisfied. If you've studied truth tables then this corresponds to the case for "$P \Rightarrow Q$" where "P" is false.

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you confused me more. For 1.), you are saying "for all elements of $X$ they are elements of $X$." Is there any doubt about it? Why this unintuitive way to explain this stuff. –  CroCo Jun 7 at 7:25
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Of course there's no doubt about it; that's why $X\subseteq X$ :p The symbol "$\subseteq$" has been defined so that it makes sense to ask for any $Y$ whether $Y\subseteq X$. Hence we can ask about $X$ itself; the answer turns out to be pretty trivial, but sometimes trivial truths are good to know. –  Malice Vidrine Jun 7 at 8:58
    
@MaliceVidrine, this $Y \subseteq X$ makes perfect sense to me. Also, $ X = X $; Because $ \subseteq $ implies $ X \subset X$, this is where I got confused. How come a set becomes a part of itself? –  CroCo Jun 8 at 1:24
    
The definition of $\subseteq$ has been explained. $X\subseteq X$ because of the very meaning of $\subseteq$--it's a logical truth that $X$ contains all the things that $X$ contains. –  Malice Vidrine Jun 8 at 1:54
    
@CroCo:$Y\subseteq X$ does not imply $Y\subset X.$ The notation $Y\subseteq X$ means $Y\subset X$ or $Y=X.$ In general $P$ or $Q$ does not imply $P$. All it means is that at least one of the statements $P$ or $Q$ is true. They do not both need to be true. Applying that to this situation, you cannot use $Y\subseteq X$ to conclude that $Y\subset X.$ In particular, you cannot use $X\subseteq X$ to conclude that $X\subset X.$ –  Will Orrick Jun 8 at 2:51

$X$: "Hello set $Y$, is it so that any element of you is also an element of me?"

$Y$: "Yes."

$X$: "Well let us practicize a notation that expresses this fact: $Y\subseteq X$."


If $X$ would have asked the same question to himself then the answer would have been 'yes' too, right? So in the suggested notation: $X\subseteq X$.


Concerning 2):

$X$: "Hello set $\emptyset$, is it so that any element of you is also an element of me?"

$\emptyset$: "Well, I have no elements that are not. Even stronger, I have no elements at all. So again the answer is yes"

$X$: "Well let us practicize a notation that expresses this fact: $\emptyset\subseteq X$."

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Awesome dialogue! –  5space Jun 7 at 16:07
    
@drhab, again with the voodoo. why is $X \subset X$ true? I'm not finding a good answer for this question. –  CroCo Jun 8 at 1:47
    
Read carefully; it's $X \subseteq X$, not $X \subset X$. –  moshbear Jun 8 at 9:07
    
If it is - in spite of all the answers on this question - not yet clear to you that $X\subseteq X$ then I cannot be of help anymore. The notation $A\subset B$ is quite often used to express that $A$ is a proper subset of $B$, wich means that $A\subseteq B$ and $A\neq B$. That means that $X\subset X$ is simply not true. But as I said: quite often, but not always. There are lots of authors using notation $\subset$ instead of $\subseteq$ and using notation $\subsetneq$ instead of $\subset$. In every book/script you should check on what side the author is on. –  drhab Jun 8 at 14:35

The relation $\subseteq$ is just like the relation $\le$. It is not wrong to say that $x \le x$, i.e., "$x$ is less than or equal to $x$", although it's also true that $x = x$. The first is a weaker statement than the second, in that it conveys less information. But it's certainly not false in any way. Look at it another way: $\le$ [is less than or equal to] means the same thing as $\not >$ [is not greater than]. Is it true that $x$ is not greater than $x$? Yes. Then it is true that $x$ is less than or equal to $x$.

Similarly, $\subseteq$ [is a subset of] is equivalent to saying "does not contain elements not in". For example, $X \subseteq Y$ means "$X$ does contain any element that is not in $Y$". Is it true that $X$ does not contain any element not in $X$? Yes, it is always true. So $X \subseteq X$.

Now, let's apply this definition to the statement $\phi \subseteq X$. We know that $\phi = \{\}$, and let $X = \{1, 2, 3\}$, as in your example. Now, is it true that $\phi$ does not contain any element not in $X$? Yes! So $\phi \subseteq X$.

In more formal notation:

  1. $X \subseteq Y \Leftrightarrow (\forall x \in X, x \in Y) \Leftrightarrow (\not\exists x \in X, x \notin Y)$
  2. $(\forall x \in X, x \in X) \Rightarrow X \subseteq X$
  3. $\not\exists x \in \phi \Rightarrow (\not\exists x \in \phi, x \notin X) \Rightarrow \phi \subseteq X$
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why it is not wrong to say $x < x$? Let x = 4, saying 4 < 4 makes no sense at all. –  CroCo Jun 8 at 1:29
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You seem to be rather confused. $4 \leq 4$ does not imply $4<4$. Obviously $3\leq 4$, but this doesn't imply $3=4$; you're making a similar mistake. –  Malice Vidrine Jun 8 at 2:46

$(1)$ Lets consider something that may seem more familiar. The symbol "$\le$" means less than on or equal to, so if we see $a\le b$, one of the following criteria hold:

$(i)\,a\lt b$ ($a$ is strictly less than $b$), or

$(ii)\, a=b$.

Now it is clear that $a\le a$ because $(ii)$ holds.

Now we will work with subsets, $Y\subseteq X$ if:

$(i)$ every element of $Y$ is an element of $X$ (but $Y\ne X$), or

$(ii)$ $Y=X$

Now it is clear that $X\subseteq X$, because $(ii)$ holds.

$(2)$ the empty set $\varnothing$ is the set consisting of no elements. It is true that for any set $X$ we have $\varnothing\subseteq X$. So let us look at this constructively, as this a good way to see how things work.

How would one create a subset of $X$? One way would be to just remove elements of $X$, then what is left over would satisfy $(i)$, and thus be a subset.

Now remove all the elements of $X$ from $X$, what are you left with? A set consisting of nothing, I.e the eempty set, but still $(i)$ holds, so we see that $\varnothing\subseteq X$

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