MHB Why Is the Empty Set Considered Unique?

  • Thread starter Thread starter evinda
  • Start date Start date
  • Tags Tags
    Relation
AI Thread Summary
The discussion centers on the uniqueness of the empty set, with participants exploring various proofs. It is established that if two sets, a and b, are both empty, they must be equal based on the axiom of extensionality, which states that two sets are equal if they contain the same elements. The conversation highlights that both sets having no elements leads to a contradiction if assumed unequal. Participants also debate the validity of using proof by contradiction versus direct proof methods. Ultimately, the consensus is that there are multiple valid approaches to demonstrate the uniqueness of the empty set.
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Cool)

Sentence

The set, that does not contain any element, is unique.

Proof:

Let's suppose that $a,b$ are sets, so that each of these sets does not contain any element and $a \neq b$.

From the axiom: Two sets, that have the same elements, are equal., there is (without loss of generality )

$$x \in a \text{ and } x \notin b (*)$$

$a,b$ do not contain any element.

$$\forall x (x \notin a)$$
$$\forall x (x \notin b)$$

$$\forall x (x \notin a \leftrightarrow x \notin b) (**)$$

From $(*)$ and $(**)$, we have a contradiction, so the set that does not contain any element is unique.Could you explain me how we get the relation $(**)$ ?
 
Physics news on Phys.org
evinda said:
$$\forall x (x \notin a)$$
$$\forall x (x \notin b)$$

$$\forall x (x \notin a \leftrightarrow x \notin b) (**)$$

...

Could you explain me how we get the relation $(**)$ ?
In general, from $\forall x\;P(x)$ and $\forall x\;Q(x)$ we can conclude $\forall x\;(P(x)\leftrightarrow Q(x))$. Indeed, for every $x$ both sides of the equivalence $P(x)\leftrightarrow Q(x)$ are true.

I wouldn't use a proof by contradiction for this. The axiom says
\[
a=b\leftrightarrow \forall x\;(x\in a\leftrightarrow x\in b)
\]
If $a$ and $b$ are empty sets, then $x\in a$ and $x\in b$ are both false for all $x$, so $\forall x\;(x\in a\leftrightarrow x\in b)$ holds, which implies $a=b$.
 
Evgeny.Makarov said:
In general, from $\forall x\;P(x)$ and $\forall x\;Q(x)$ we can conclude $\forall x\;(P(x)\leftrightarrow Q(x))$. Indeed, for every $x$ both sides of the equivalence $P(x)\leftrightarrow Q(x)$ are true.

I understand! (Smile)

I wouldn't use a proof by contradiction for this. The axiom says
\[
a=b\leftrightarrow \forall x\;(x\in a\leftrightarrow x\in b)
\]

Evgeny.Makarov said:
If $a$ and $b$ are empty sets, then $x\in a$ and $x\in b$ are both false for all $x$, so $\forall x\;(x\in a\leftrightarrow x\in b)$ holds, which implies $a=b$.

Could you explain it further to me? (Sweating)
 
Hmm, I am not sure what to explain. Which part is not clear?
 
Evgeny.Makarov said:
Hmm, I am not sure what to explain. Which part is not clear?

I haven't understood why this: $\forall x (x \in a \leftrightarrow x \in b)$ holds, although $x \in a$ and $x \in b$ are both false..
Shouldn't it be $\forall x (x \notin a \leftrightarrow x \notin b)$ ? Or am I wrong? (Thinking)
 
$A\leftrightarrow B$ holds iff both $A$ and $B$ are true or if both of them are false.
 
Evgeny.Makarov said:
If $a$ and $b$ are empty sets, then $x\in a$ and $x\in b$ are both false for all $x$, so $\forall x\;(x\in a\leftrightarrow x\in b)$ holds, which implies $a=b$

Evgeny.Makarov said:
$A\leftrightarrow B$ holds iff both $A$ and $B$ are true or if both of them are false.

A ok.. I got it! (Nod)

Is the proof by contradiction wrong? (Thinking)
 
evinda said:
Is the proof by contradiction wrong?
I wrote about it in https://driven2services.com/staging/mh/index.php?posts/58821/.
 
From the axiom: Two sets, that have the same elements, are equal., there is (without loss of generality )

$$x \in a \text{ and } x \notin b (*)$$

Could you explain me why the above is the contraposition of the axiom of extension? (Thinking)
 
  • #10
evinda said:
Could you explain me why the above is the contraposition of the axiom of extension? (Thinking)

By the axiom of extensionality we have:

$$a=b\Longleftrightarrow\forall x[x\in a\Longleftrightarrow x\in b]$$

THAT IMPLIES:

$$a=b\Longleftarrow\forall x[x\in a\Longleftrightarrow x\in b]$$

AND by contrapositive law we have:

$$a\neq b\Longrightarrow\neg[\forall x(x\in a\Longleftrightarrow x\in b)]$$

BUT

$$\neg[\forall x(x\in a\Longleftrightarrow x\in b)]$$ is equevalent to:

$$\exists x[ \neg((x\in a\Longrightarrow x\in b)\wedge(x\in b\longrightarrow x\in a))]$$

AND that implies ,by using D MORGAN$$ \neg(x\in a\Longrightarrow x\in b)\vee\neg(x\in b\longrightarrow x\in a)$$

Which is equivalent to:

$$(x\in a\wedge\neg x\in b)\vee(x\in b\wedge\neg x\in a)$$
 
  • #11
evinda said:
Hello! (Cool)

Sentence

The set, that does not contain any element, is unique.

Proof:

Let's suppose that $a,b$ are sets, so that each of these sets does not contain any element and $a \neq b$.

From the axiom: Two sets, that have the same elements, are equal., there is (without loss of generality )

$$x \in a \text{ and } x \notin b (*)$$

$a,b$ do not contain any element.

$$\forall x (x \notin a)$$
$$\forall x (x \notin b)$$

$$\forall x (x \notin a \leftrightarrow x \notin b) (**)$$

From $(*)$ and $(**)$, we have a contradiction, so the set that does not contain any element is unique.Could you explain me how we get the relation $(**)$ ?

A proof by contradiction is the following:

Let $$a\neq b$$

Now:

suppose $$\neg x\in a$$.......1

Since b is empty we have :

$$\forall x (\neg x\in b)$$
OR

$$ (\neg x\in b)$$........2

And by the rule of conditional proof we can conclude:

$$\neg x\in a\Longrightarrow\neg x\in b$$........3...

In the same way we infer that:

$$\neg x\in b\Longrightarrow\neg x\in a $$.................4

Now from (3) and using contrapositive we have:

$$x\in b\Longrightarrow x\in a$$................5

In same way;

$$x\in a\Longrightarrow x\in b$$.................6

FROM (5) and (6) we can conclude:

$$x\in a\longleftrightarrow x\in b$$
OR

$$\forall x[x\in a\Longleftrightarrow x\in b]$$

And by the axiom of extensionality we have: a=b a contradiction since we assumed $$a\neq b$$

Hence : a=b

There at least 5,6 different ways one can prove the uniqueness of the empty set
 

Similar threads

Back
Top