MHB Is A the Empty Set When A is a Subset of the Empty Set?

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The discussion centers on proving that if set A is a subset of the empty set (A ⊆ ∅), then A must also be the empty set (A = ∅). The reasoning provided establishes that if A were not empty, it would contain an element that contradicts its subset relationship with the empty set. Participants confirm the validity of the proof and explore alternative ways to express the relationship between A and the empty set. The conversation emphasizes the logical implications of set theory, particularly the axiom of extensionality. Overall, the conclusion is that A must indeed be the empty set if it is a subset of the empty set.
evinda
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Hello! (Wave)

I want to show that $A \subset \varnothing \rightarrow A=\varnothing$.

That's what I thought:

$$A \subset \varnothing \text{ means that :}$$
$$\forall x (x \in A \rightarrow x \in \varnothing)$$

Since, there is no $x$, such that $x \in \varnothing$, there is no $x$, such that $x \in A$.

Therefore, $A$ is the empty set.

Could you tell me if it is right? (Thinking)
 
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Well, we need to prove that if $ A \subset \emptyset$ it follows that $A = \emptyset$.

Proof
Suppose $A \neq \emptyset$ then we can find an $x \in A$ such that $x \notin \emptyset$. Since $A \subset \emptyset$ it follows that $x \notin A$ hence we have a contradiction. Therefore our assumption was false and thus $A = \emptyset$.
 
evinda said:
$$A \subset \varnothing \text{ means that :}$$
$$\forall x (x \in A \rightarrow x \in \varnothing)$$

Since, there is no $x$, such that $x \in \varnothing$, there is no $x$, such that $x \in A$.

Therefore, $A$ is the empty set.

Could you tell me if it is right?
Yes, you are right.
 
Siron said:
Well, we need to prove that if $ A \subset \emptyset$ it follows that $A = \emptyset$.

Proof
Suppose $A \neq \emptyset$ then we can find an $x \in A$ such that $x \notin \emptyset$. Since $A \subset \emptyset$ it follows that $x \notin A$ hence we have a contradiction. Therefore our assumption was false and thus $A = \emptyset$.

Evgeny.Makarov said:
Yes, you are right.

Nice, thank you very much! (Smile)
 
Evgeny.Makarov said:
Yes, you are right.

Could we also say it like that? (Thinking)

It is known that $\varnothing \subset B$, for all sets $B$.

Therefore, $\varnothing \subset A$.
We also now that $A \subset \varnothing$.

So, we conclude that $A=\varnothing$.
 
evinda said:
We also now that $A \subset \varnothing$.
We now... what? The suspense is killing me! This reminds me of the "I Accidentally…" meme. (Smile)

If you mean that we assumed that $A\subset\varnothing$ (I prefer to use $\subseteq$ for possibly nonstrict inclusions), then you are correct. The only thing is you have to show that $A\subseteq B$ and $B\subseteq A$ imply $A=B$. This is obvious, but so is the original statement, so it's difficult to say whether its proof is required.
 
Evgeny.Makarov said:
We now... what? The suspense is killing me! This reminds me of the "I Accidentally…" meme. (Smile)

(Giggle)

Evgeny.Makarov said:
If you mean that we assumed that $A\subset\varnothing$ (I prefer to use $\subseteq$ for possibly nonstrict inclusions), then you are correct. The only thing is you have to show that $A\subseteq B$ and $B\subseteq A$ imply $A=B$. This is obvious, but so is the original statement, so it's difficult to say whether its proof is required.

Could prove it like that? (Thinking)

$A \subseteq \varnothing \leftrightarrow \forall x(x \in A \rightarrow x \in \varnothing)$

$\varnothing \subseteq A \leftrightarrow \forall x(x \in \varnothing \rightarrow x \in A)$

$A \subseteq \varnothing \wedge \varnothing \subseteq A \leftrightarrow \forall x(x \in A \leftrightarrow x \in \varnothing) \overset{\text{ axiom of extensionality }}{ \rightarrow } A=\varnothing$
 
Yes, this is perfect.
 
Evgeny.Makarov said:
Yes, this is perfect.

Nice! Thanks a lot! (Happy)
 
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