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I know that sometimes In set theory, $\emptyset \subseteq A$ is true, whereas $\emptyset \in A$ is false. What is the difference between $\emptyset \subseteq A$ and $\emptyset \in A$ ?

User
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  • Everything in an empty bag is also in a nonempty bag, but a nonempty bag need not contain an empty bag. Do you understand the difference between $\subseteq$ and $\in$? – anon May 06 '15 at 05:25
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    $\varnothing \subset {1}$ but $\varnothing \notin {1}$ – Jose Antonio May 06 '15 at 05:28
  • I said $\subseteq$ and $\in$, not $\emptyset$ and $\in$. Visualize my comment with grocery bags. You can put bags inside bags, and sets work the same way. One set being an element of another is not the same as one set being a subset of another. Either you see the difference, or you don't... in any case, set theory is not designed to cater to our physical intuitions, it is pure logic, so you have to face it with an eye for only pure logic. – anon May 06 '15 at 05:31
  • Sorry, I feel like I am just a hair away from understanding, but not quite there. @Jose Antonio, can you elaborate on why that is the way it is? – User May 06 '15 at 05:36
  • Everything in an empty bag is also in a bag containing an apple. But a bag containing an apple does not itself have an empty bag inside it. That is Jose's example, except with an apple instead of the number $1$. – anon May 06 '15 at 05:37
  • The empty set is always subset of any set. We say $A\subset B$ iff all elements in $A$ are also in $B$, in particular this is vacuously true when $A=\varnothing$; the only way in which this were false is if we can show an element in $\varnothing$ which is not in $B$ (where $B$ is any set) but this is impossible, $\varnothing$ does not have elements!!!. On the other hand $a\in B$, means that $a$ is an element of $B$, try to understand why $\varnothing \notin {{\varnothing}}$, but $\varnothing \subset {{\varnothing}}$ – Jose Antonio May 06 '15 at 05:41
  • also note that $\varnothing \in {\varnothing}$ and $\varnothing \subset {\varnothing}$ – Jose Antonio May 06 '15 at 05:43
  • $\varnothing \notin {{\varnothing}}$ because the only element is ${\varnothing}$ and this is not empty, contains one element in it. Maybe the following can help you, is not the same nothing than a box containing nothing (this illustrates the difference of $\varnothing$ and ${\varnothing}$) and in the example we have a box containing a box (even though this contains nothing) and this box is not empty because contains a box – Jose Antonio May 06 '15 at 05:58
  • @Jose Antonio, thank you, I think I understand now. And thank you to everybody else who contributed. Really appreciate it. – User May 06 '15 at 06:07
  • see also here – user 1 May 06 '15 at 07:42

2 Answers2

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$A \subseteq B \implies $ All elements of $A$ are in $B$.

$A \in B \implies$ $A$ is an element of $B$.

A real life example:

$\text{Trucks} \subseteq \text{Vehicles}$

$\text{Ford F150} \in \text{Trucks}.$

But,

$\text{Trucks} \not\in \text{Vehicles}$

$\text{Ford F150} \not\subseteq \text{Trucks}.$

MathMajor
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  • This makes partial sense, but the part I get confused on is $\text{Trucks} \not\in \text{Vehicles}$ and $\text{Ford F150} \not\subseteq \text{Trucks}.$ Wouldn't trucks also be an element of vehicles, and Ford F150 be a subset of Trucks? – User May 06 '15 at 05:42
  • @Omar $\rm Trucks$ is the set of all trucks. That's not even a single physical object. Can you turn the key and drive the entire set of all trucks yourself? And MathMajor here is using $\rm FordF150$ to refer to a single Ford F150, which is not a set. – anon May 06 '15 at 05:43
  • Perhaps the names are vague. I meant $\text{Vehicles} = { \text{Cars, Trucks, Planes}, \dots}$ and $\text{Trucks} = { \text{Ford F150, Ford F250, Chevrolet Silverado}, \dots }$. – MathMajor May 06 '15 at 05:43
  • however, {Ford F150} $\subseteq$ Trucks – MichaelChirico May 06 '15 at 12:38
  • @MichaelChirico Yes. – MathMajor May 06 '15 at 23:52
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The first one is a subset and being a subset means it has to satisfy the following requirement: Suppose A $\subset$ B this means for all elements of A they must be in B, whereas A $\in$ B means that the element A is in the set B. For the first question the empty set doesn't have any elements so it vacuously satisfy being a subset for any of your sets.

For example {1} $\subset$ of {1,{1},{{1}}} since the element 1 is in that set. Now {{1}} $\in$ {1,{{1}}} since it {{1}} is an element of that set. Hope this clears the confusion !.