Why Is the Empty Set Considered Unique?

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In summary: But the simplest is the following:We can prove by using the axiom of extensionality, that the empty set is unique by showing that there is only one set satisfying the defining property of the empty set.The defining property of the empty set is that it has no elements. So we can prove that if a set satisfies the condition of having no elements, then it is equal to the empty set.\begin{itemize} \item We know that the empty set has no elements. So we can say $\forall x(x \notin a)$ \item If $a$ satisfies the condition of having no elements, then $x \notin a$ is always true. So we can say $\forall x(x \notin a \leftrightarrow
  • #1
evinda
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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 $(**)$ ?
 
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  • #2
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 $a=b$.
 
  • #3
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)
 
  • #4
Hmm, I am not sure what to explain. Which part is not clear?
 
  • #5
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)
 
  • #6
$A\leftrightarrow B$ holds iff both $A$ and $B$ are true or if both of them are false.
 
  • #7
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)
 
  • #8
evinda said:
Is the proof by contradiction wrong?
I wrote about it in https://driven2services.com/staging/mh/index.php?posts/58821/.
 
  • #9
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:

\(\displaystyle a=b\Longleftrightarrow\forall x[x\in a\Longleftrightarrow x\in b]\)

THAT IMPLIES:

\(\displaystyle a=b\Longleftarrow\forall x[x\in a\Longleftrightarrow x\in b]\)

AND by contrapositive law we have:

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

BUT

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

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

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

Which is equivalent to:

\(\displaystyle (x\in a\wedge\neg x\in b)\vee(x\in b\wedge\neg x\in a)\)
 
  • #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 \(\displaystyle a\neq b\)

Now:

suppose \(\displaystyle \neg x\in a\).......1

Since b is empty we have :

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

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

And by the rule of conditional proof we can conclude:

\(\displaystyle \neg x\in a\Longrightarrow\neg x\in b\)........3...

In the same way we infer that:

\(\displaystyle \neg x\in b\Longrightarrow\neg x\in a \).................4

Now from (3) and using contrapositive we have:

\(\displaystyle x\in b\Longrightarrow x\in a\)................5

In same way;

\(\displaystyle x\in a\Longrightarrow x\in b\).................6

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

\(\displaystyle x\in a\longleftrightarrow x\in b\)
OR

\(\displaystyle \forall x[x\in a\Longleftrightarrow x\in b]\)

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

Hence : a=b

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

FAQ: Why Is the Empty Set Considered Unique?

How do we determine the cause and effect relationship between variables?

To determine the cause and effect relationship between variables, we must conduct a controlled experiment where we manipulate one variable (the independent variable) and observe the effect on another variable (the dependent variable). By keeping all other variables constant, we can determine if there is a causal relationship between the variables.

How do we account for confounding variables in establishing a relationship?

Confounding variables are other factors that may influence the relationship between the variables being studied. To account for these variables, researchers must carefully control or measure them in their study design. This can involve randomization, matching participants, or statistical techniques such as regression analysis.

Can correlation be used to determine a cause and effect relationship?

No, correlation alone cannot determine a cause and effect relationship. While a strong correlation between two variables may suggest a relationship, it does not prove causation. Other factors must be considered and controlled for in order to establish causation.

What is the difference between a positive and a negative correlation?

A positive correlation exists when two variables increase or decrease together. In other words, as one variable increases, the other also increases. A negative correlation exists when two variables move in opposite directions, meaning as one variable increases, the other decreases.

How do we determine the strength of a relationship between two variables?

The strength of a relationship between two variables can be determined by calculating the correlation coefficient, which is a measure of how closely the two variables are related. The correlation coefficient ranges from -1 to +1, with higher absolute values indicating a stronger relationship.

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