How would one denote a set that contains all elements in all subsets of another set

elementary-set-theorynotation

Just say i have a finite set $A=\{a,b,\{c,d\}\}$

Naturally the power set of A would be $\{\{c,d\}\}$.

But I wonder how I would denote a set, say $B$ containing all elements of all subsets of $A$, so $A$ (some function)$=\{c,d\}$ and not $\{\{c,d\}\}$. Many thanks

Best Answer

If $A=\{a,b,\{c,d\}\}$, then the power set would be the set of all subsets of $A$, so it would be: $$\mathcal P(A)=\{\varnothing,\{a\},\{b\},\{\{c,d\}\},\{a,b\},\{a,\{c,d\}\},\{b,\{c,d\}\},\{a,b,\{c,d\}\}\}$$ I'm not sure how you arrived at $\{\{c,d\}\}$.

The set containing all elements of all subsets of $A$ is then $\bigcup \mathcal P(A)$: the union of all the subsets of $A$. Of course, if a set is a subset of $A$, then its elements are elements of $A$, so the set of all elements of subsets of $A$ is just $A$ itself. That is, $\bigcup\mathcal P(A)=A$.

If instead you want the elements of the sets in $A$, then this is $\bigcup A=\{c,d\}\cup a\cup b$, or in words, the set containing $c$ and $d$, and furthermore all the sets that are elements of $a$ or of $b$.

Just to be sure that we're on the same page, here is the definition of a subset:

A subset $X$ of $A$ is a set such that all the elements of $X$ are elements of $A$. We write $X\subseteq A$. The set of all subsets of a set $A$ is called the power set and is denoted by $\mathcal P(A)$

For good measure, here is also the definition of union:

If $A$ is a set, then the union of $A$ is the set containing all the elements of the sets that are in $A$. We denote this by $\bigcup A$. If $A$ and $B$ are two sets then we denote $\bigcup\{A,B\}=A\cup B$ and confusingly also call this the union of $A$ and $B$ (instead of the union of the set containing $A$ and $B$).

Finally, note that in set theory, usually, all the objects are sets. So the things called $a$, $b$, $c$ and $d$ in your set $A$ would all be sets.

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