MHB Why can it consist of only one element of a?

[evinda]

Hey! (Nerd)

If $a,b$ sets, then we symbolize $\{ a,b\}$, the set that has as elements $a$ and $b$ and only these.

If $\{ a, b \}$ a set of pair and $a=b$, then we write $\{a,a \}=\{a \}$ and the set $\{ a \}$ is called the singleton, that consists of only one element of $a$.

Why can $\{a\}$ consist of only one element of $a$ ? (Thinking)

 

[Evgeny.Makarov]

Why can $\{a\}$ consist of only one element of $a$ ?

Strictly speaking, it is not the case that $\{a\}$ consists of only one element of $a$; rather, $\{a\}$ consists of only one element, namely, $a$.

By definition of $\{a,b\}$, we have
\[
x\in \{a,b\}\leftrightarrow x=a\lor x=b.\qquad(*)
\]
Also by definition, $\{a\}=\{a,a\}$. Instantiating (*) with $\{a\}$ (i.e., when $a=b$), we get
\[
x\in \{a\}\leftrightarrow x=a\lor x=a\leftrightarrow x=a.
\]

 

[evinda]

Strictly speaking, it is not the case that $\{a\}$ consists of only one element of $a$; rather, $\{a\}$ consists of only one element, namely, $a$.

By definition of $\{a,b\}$, we have
\[
x\in \{a,b\}\leftrightarrow x=a\lor x=b.\qquad(*)
\]
Also by definition, $\{a\}=\{a,a\}$. Instantiating (*) with $\{a\}$ (i.e., when $a=b$), we get
\[
x\in \{a\}\leftrightarrow x=a\lor x=a\leftrightarrow x=a.
\]

I understand... Thank you very much! (Clapping)

 

[Deveno]

Note that:

$P \vee P \iff P$

"If I am going to the store, or, I'm going to the store, then, I'm going to the store. Moreover, if I'm going to the store, then surely, either I'm going to the store, or...I'm going to the store!"

Note as well, that:

$\{a,a\} = \{a\} \cup \{a\}$

More pointedly: "sets are not multisets", or as G. Spenser-Brown says, in The Laws of Form (paraphrased):

To call again, is to call.

 

[mathbalarka]

This is actually known as the axiom of extensionality. It is defined that two sets $A$ and $B$ are equal if every element of $A$ is in $B$. From this follows that $\{1, 1\}$ for example is equal to the singleton set $\{1\}$.

However, this is only an axiom. There are set-like structures for which this axiom doesn't hold, e.g., $\{1, 1\}$ and $\{1\}$ differs. A multiset is an example of such a structure, for example.

EDIT : Ah, Deveno beat me to it.

 

[Deveno]

This is actually known as the axiom of extensionality. It is defined that two sets $A$ and $B$ are equal if every element of $A$ is in $B$. From this follows that $\{1, 1\}$ for example is equal to the singleton set $\{1\}$.

However, this is only an axiom. There are set-like structures for which this axiom doesn't hold, e.g., $\{1, 1\}$ and $\{1\}$ differs. A multiset is an example of such a structure, for example.

EDIT : Ah, Deveno beat me to it.

Do we get extra credit for thinking of the same thing, at the same time?

 

[mathbalarka]

Do we get extra credit for thinking of the same thing, at the same time?

Hahaha, probably not. We'll be marked as copy-cats by the examiners soon enough.