Group the following into sets that are equal.
(a) $\emptyset$
(b) $\{\emptyset\}$
(c) $\{0\}$
(d) $\{\{\}\}$
(e) $\{\}$
I know $\emptyset$ is a subset of all empty sets. Thus, would these not all be equal to each other?
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$\begingroup$A set can be thought of in a physical sense as a bag with possibly some things inside. It is possible that inside of this bag are numbers, objects, names, or maybe even other bags (which then could possibly have things inside of them).
Two sets are said to be equal if they both are representations of the same thing. Although $\emptyset$ is a subset of all other sets, there are things which are "bigger than" the empty set and are therefore not equal to it.
The "bag" then is displayed with a single pair of brackets denoting where the bag begins and where it ends. For example the set $\{1,2,3\}$ is like a "bag with a 1 a 2 and a 3 inside of it"
In this way of thinking, the empty set is like a bag with nothing in it. It can be displayed as $\{~\}$ or we also give it its own special symbol so that we can display it more cleanly as $\emptyset$. Some important sets that we use frequently we like to give special symbols that we reserve solely for their own special use, for example $\Bbb N, \Bbb Q, \Bbb C$.
The set $\{1\}$ is then like a bag with a 1 inside of it.
The set $\{~\{1\}~\}$ however is a bag with another bag inside of it where that inside bag has a 1 in it.
$$\underbrace{\{~~~~\overbrace{\{~~1~~\}}^{\text{inside bag}}~~~~\}}_{\text{outside bag}}$$
What does $\{~\{~\}~\}$ represent then if you think of it in terms of bags and items within bags?
It is a bag with an empty bag inside of it
Is this the same thing as $\{~\}$?
No, $\{~\}$ only has one bag and it is empty, whereas $\{~\{~\}~\}$ has two bags, one inside of the other.
How about $\{\emptyset\}$?
We said that $\emptyset$ is an empty bag, so $\{\emptyset\}$ is an $\emptyset$ inside of a bag. In other words, a bag with an empty bag inside of it.
How about $\{0\}$?
This is a bag with a zero inside of it. None of the other sets in our question have any zeroes or ones or any other numbers...
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$\endgroup$ 3 $\begingroup$Okay... if you want to be really picky, It technically depends on what you mean by "$0$." Under most interpretations, $0\neq \emptyset$ and these are two completely different objects, but in some cases it is useful to define the natural numbers as sets in the first place in which case...you could interpret them as being the same. I would not.
Your post has been edited. In the new edit it is clear that you are not doing the confusing and abstract and advanced set theory notation of numbers whereas before it seemed clear you were. SO new answer:
"I know ∅ is a subset of all empty sets. Thus, would these not all be equal to each other?"
No, having something as an element is not the same thing as being a subset. I hope this will become clear in the explanation. (If not, you can use the "a set is like a bag" analogy and "an empty bag inside a bag is different than an empty bag")
a) $\emptyset$ is the empty set. It is a set with no elements whatsoever. Nothing in it.
b) {$\emptyset$} is a set that has one element. The one element is the empty set. It's okay to put sets as elements of sets. But the sets are elements of the set. They are not subsets. A subset is a set with some of the same elements. A set inside a set may have entirely different elements of its own. At any rate it is in the first set as an element.
c) {0} is a set with the single element 0 in it. Zero is a number. That's not a set.
So for $a \ne b$, $b \ne c$ and $c \ne a$.
d) {} is another way of writing the empty set. The "{" and "}" represent symbols to indicate "what's inside are the elements of the set". So {} says "this is a list with nothing in it. That's the empty set.
So {{}} is the set that has a single element. That element is the empty set.
So we have a=$\emptyset$ = {}. b = d = {$\emptyset$} = {{}}. And c = {0}.
e) {} as I stated in d) {} is the empty set.
So
a = e = the empty set = a set with no elements at all = $\emptyset$ = {}
b =d = a set with the empty set as its one and only element = {{}} = {$\emptyset$}
c = a set with the number zero as its one and only element = {0}.
==== old answer assuming you are doing a way more advanced class. Left for reference.=====
a) If you are not doing the set theoretical definition of number, 0 is not a set at all. It's just a number.
If you are doing the set theoretical definition of number, everything is a set. I'm going to assume you are doing the set theoretical definition of number for the rest of this answer.
0 is the empty set-- a set with no elements. There is only one empty set so saying "0 is a subset of all empty sets" is misleading. 0 is the empty set--- the one and only empty set.
b) {0}. This is a set with an element in it. The element in it is 0. This is not the empty set because it has an element in it. We can rewrite the empty set as, $\emptyset$, {}, or 0. So this set can be written as {0}, {{}}, {$\emptyset$} or 1. As in the set definition of numbers the number 1 is defined to be the set that contains just one member-- the empty set.
c) is the same as b in writing. Was one of these supposed to be {$\emptyset$}? Well, whatever. As I explained in b) {{}}, {0}, {$\emptyset$} and 1, are all the same thing: a set with exactly one element; the element is the empty set.
d) ditto
e) {} is a set with no elements-- i.e. the empty set.
So a = e and b=c=d.
Being an element of is not the same thing as being a subset of.
====
Okay, if you are not doing the set theory definition of numbers, then 0 is a number. It is not a set at all. Otherwise everything is the same: a) not a set b=c=d= the set containing the empty set. e= the empty set.
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