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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 $\emptyset \in A$ ?

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  • $\begingroup$ 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 $\in$? $\endgroup$ – anon May 6 '15 at 5:25
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    $\begingroup$ $\varnothing \subset \{1\}$ but $\varnothing \notin \{1\}$ $\endgroup$ – Jose Antonio May 6 '15 at 5:28
  • $\begingroup$ 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. $\endgroup$ – anon May 6 '15 at 5:31
  • $\begingroup$ 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? $\endgroup$ – Omar N May 6 '15 at 5:36
  • $\begingroup$ 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 $1$. $\endgroup$ – anon May 6 '15 at 5:37
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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,

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

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

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  • $\begingroup$ 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? $\endgroup$ – Omar N May 6 '15 at 5:42
  • $\begingroup$ @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. $\endgroup$ – anon May 6 '15 at 5:43
  • $\begingroup$ Perhaps the names are vague. I meant $\text{Vehicles} = \{ \text{Cars, Trucks, Planes}, \dots\}$ and $\text{Trucks} = \{ \text{Ford F150, Ford F250, Chevrolet Silverado}, \dots \}$. $\endgroup$ – MathMajor May 6 '15 at 5:43
  • $\begingroup$ however, {Ford F150} $\subseteq$ Trucks $\endgroup$ – MichaelChirico May 6 '15 at 12:38
  • $\begingroup$ @MichaelChirico Yes. $\endgroup$ – MathMajor May 6 '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 !.

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