Set theory with empty sets confuses me. Why is {Ø} ⊂ {{Ø}} false and why is {{Ø}} ⊂ {Ø, {Ø}} true?

Question

I get that $\varnothing$ means the same as $\{\}$, it's still confusing to me. The above statements and their validity come from my University mathematics handbook on Discrete Mathematics.

The first statements makes it seem like $\{\varnothing\}$ and $\{\{\varnothing\}\}$ are the same thing. Then the second statement immediately dispels it by saying that $\{\{\varnothing\}\}$ isn't the same thing as $\{\varnothing\}$.

I want to mention that they make a distinction between $\subset$ and $\subseteq$.

• {A} ⊂ {A} would be false
• {A} ⊂ {{A}, B} would be correct
• {A} ⊆ {A} would be true
• A ⊂ {A} is correct too I think because {A} is a singleton and is distinct from just A.

Answer 1

$$\color{red}{\{\emptyset\}}\subset \{\color{red}{\{\emptyset\}}\}$$

$$\{ \color{green}{\{\emptyset\}}\}\subset \{\emptyset,\color{green}{\{\emptyset\}}\}$$

The first ( colored red) is trying to say an element is a subset . The one colored green, has a set with a single element which is a set with the emptyset as an element. Which so happens to be an element in the right hand set.

Answer 2

Keep in mind, that a subset is a set whose elements (individual pieces of the set) are all elements of another

The empty set is given by

$$\emptyset = \lbrace \rbrace$$ as you have identified. However, it is important to recognize that

$$ \lbrace \emptyset \rbrace = \lbrace ~\lbrace \rbrace~ \rbrace $$

is not a subset of $$ \lbrace \lbrace \emptyset \rbrace \rbrace = \lbrace ~\lbrace ~\lbrace \rbrace ~\rbrace ~\rbrace $$

Because $ \lbrace \rbrace$ on it's own is not a element of $\lbrace \lbrace \emptyset \rbrace \rbrace$, and $\lbrace \rbrace$ is the only element of $\lbrace \emptyset \rbrace$

The second example given IS true, becuase $\lbrace \emptyset \rbrace$ is the only element of $\lbrace \lbrace \emptyset \rbrace \rbrace$ and also an element of $\lbrace \emptyset , \lbrace \emptyset \rbrace \rbrace$

Answer 3

The last of your four bullet points is not correct. It would be true to say that $A\in \{A\}$, but not $A\subset \{A\}$.

The former statement means that the object (not necessarily a set) on the left is one of the top-level things in the set on the right, i.e. if you remove the outer braces from the set on the right you get a list, and the object on the left appears in that list.

The latter statement is saying that the set on the left consists of elements of the set on the right (and not all of them). In other words, if you remove outer braces from both sides you get two lists, and you need everything in the first list to appear in the second list (but the lists shouldn't be identical).

Now $\{\varnothing\}\subset\{\{\varnothing\}\}$ is not true, because the only element of the left-hand side is $\varnothing$, whereas the only element of the right-hand side is $\{\varnothing\}$, and $\varnothing\neq\{\varnothing\}$.

However, $\{\{\varnothing\}\}\subset\{\varnothing,\{\varnothing\}\}$ is true because the right-hand side has two elements $\varnothing$ and $\{\varnothing\}$, and the only element of the left-hand side is $\{\varnothing\}$, which is one of them.

Answer 4

There are two answers, but I want to answer too, bocouse there is a simple way to look at this "problem".

Becouse $\emptyset$ is an empty set (has no elments) and $\{\emptyset\}$ is not an empyt, becouse it has one element: the set (no matter what the set "has inside") then for $A=\{\emptyset\}$ and $B=\{\{\emptyset\}\}$ and for $a=\emptyset$, $b=A=\{\emptyset\}$ we have $$A=\{a\}\\ B=\{b\}=\{A\}$$ so is not true, that all elements from $A$ are in $B$, becouse $B$ don't contain $a$.

But, if $C=\{\emptyset, \{\emptyset\}\}=\{a,b\}$ then $B \subset C$, becouse the only element of $B$ (that is $b=A=\{\emptyset\}$) is also contained by $C$.