Is $\varnothing\in\{\varnothing\}$ and $\varnothing\subseteq \{\varnothing\}?$

Question

As per me yes they are true because, in the first statement, the set $\{\varnothing\}$ contains one single element, $\varnothing$, the empty set.

In the second statement because of the fact that an empty set is a subset of every set

What about $\{\varnothing\}\subseteq \varnothing$ ?

As you already said: ${\emptyset}$ contains $1$ element. $\emptyset$ contains no elements. — Quotenbanane, Sep 22 '21 at 18:18
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score 3 · Answer 1 · answered Sep 22 '21 at 18:24

3

The only subset of $\emptyset$ is $\emptyset$. If $A\subseteq\emptyset$ and $A$ has an element $a$, then $a$ must also be an element of $\emptyset$.

answered Sep 22 '21 at 18:24

Francis Adams

2,808

Cool Thank You – Ayush Sambher Sep 22 '21 at 18:26

score 2 · Answer 2 · answered Sep 22 '21 at 18:18

2

The empty set is a subset of every set. Hence $\varnothing \subseteq \{\varnothing\}=:S$. The only element of $S$ is $\varnothing$ by inspection. Hence $\varnothing\in S$. Note that $S$ has one element whereas $\varnothing$ has none; hence $S\not\subseteq \varnothing$, since, otherwise, $S=\varnothing$.

answered Sep 22 '21 at 18:18

Shaun

44,997

1

Got it thanks – Ayush Sambher Sep 22 '21 at 18:26

score 2 · Answer 3 · answered Sep 22 '21 at 18:27

2

Use $\{\}$ instead of $\emptyset$ and everything will become clear.

$\{\} \in \{\{\}\}$ true
$\{\} \in \{\}$ false
$\{\} \subseteq \{\}$ true
$\{\} \subseteq \{\{\}\}$ true
$\{\{\}\} \subseteq \{\}$ false

answered Sep 22 '21 at 18:27

Kolja

2,760

Indeed! Thank You – Ayush Sambher Sep 22 '21 at 19:27

Joe · Accepted Answer · 2022-03-03T11:08:36.927

Yes, by definition $\varnothing\in\{\varnothing\}$—the notation $\{\varnothing\}$ represents the set that contains the empty set (and no other element). And it is also true that $\varnothing\subseteq\{\varnothing\}$. This is because $A\subseteq B$ is just an abbreviation of $$ \forall x:x\in A\implies x\in B \, . $$ Therefore, the claim that $\varnothing\subseteq\{\varnothing\}$, by definition means $$ \forall x:x\in\varnothing\implies x\in\{\varnothing\} \, . $$ Recall that an implication $p\implies q$ is always satisfied when $p$ is false (see here). Since the empty set is, well, empty, there is no value of $x$ for which $x\in\varnothing$ is true. Hence, the claim$$\forall x:x\in\varnothing\implies x\in\{\varnothing\}$$ follows immediately. This is known as a vacuous truth. Note that exactly the same argument can be used to prove that the empty set is a subset of any given set.

As for $\{\varnothing\}\subseteq \varnothing$, this claim is not true. To prove this, consider that the negation of $$\forall x:x\in\{\varnothing\}\implies x\in\varnothing$$ is $$\exists x:x\in\{\varnothing\}\land x\notin\varnothing \, .$$The above statement is true because $\varnothing\in\{\varnothing\}$ and $\varnothing\not\in\varnothing$.

No worries, @AyushSambher! Please let me know if you have any questions. — Joe, Sep 22 '21 at 19:53

Is $\varnothing\in\{\varnothing\}$ and $\varnothing\subseteq \{\varnothing\}?$

4 Answers4