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Can someone explain me this: $2^{\{1,2\}}$ I know this equals to: $\{\varnothing, \{1\}, \{2\}, \{1, 2\}\}$ Right?

So $\{1\}$ for example is an element of $2^{\{1,2\}}$, $\{1\} \in 2^{\{1,2\}}$

But can someone explain me to what this notation equals? $2^{\{1,2,3\}}$

Does it equal $\{\varnothing, \{1\}, \{2\}, \{1, 2, 3\}\}$?

And why is $\{\{1\}, \{2\}\}$ a subset of $2^{\{1,2,3\}}$?

Thanks! I'm very new to math at my university and I lacked the basics. But I'm studying to catch up!

N. F. Taussig
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O'Niel
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  • Do you know what subset means? $2^X$ is the set of subsets of $X$, see power set. – M. Winter Sep 29 '17 at 16:26
  • Also there is an explanation/codification with binary triples. –  Sep 29 '17 at 16:34
  • To understand why it's $2$ (rather, why a finite set $X$ with cardinality $n$ has a power set with cardinality $2^n$), you pick a subset of $X$ by going through each element of $X$ and choosing one of two options: either the element is in your subset or not. So $n$ times, you have two options, and each choice is independent from the others, so you have $2^n$ possible subsets. – Kevin Long Sep 29 '17 at 21:09

3 Answers3

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You're talking about powersets. A powerset of a set is the set of all possible subsets of that set. The reason $2^S$ is used is because the number of elements in the powerset is $2$ raised to the power of the number of elements in $S$.

$2^{\{1,2,3\}}$ contains all $8$ subsets of $\{1,2,3\}$:
$\{\}$
$\{1\}$
$\{2\}$
$\{3\}$
$\{1,2\}$
$\{1,3\}$
$\{2,3\}$
$\{1,2,3\}$

Note that each element of $2^{\{1,2,3\}}$ is a set. As you can see from the above list of the elements of $2^{\{1,2,3\}}$, two of the sets contained in the powerset are $\{1\}$ and $\{2\}$, so those two sets taken together are indeed a subset of $2^{\{1,2,3\}}$.

Joffan
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This (confusing) notation is: $2^X$ means "all subsets of $X$". So $$ 2^{\{1,2,3\}} = \{ \\ \emptyset, \\ \{1\},\{2\},\{3\}, \\ \{1,2\},\{1,3\},\{2,3\}, \\ \{1,2,3\} \\ \} $$

GEdgar
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The notation $[n]=\{ 1,2, \cdots ,n \}$ denotes the set of the first $n$ natural numbers. (In particular $[3]= \{1,2,3\}$.) This is quite natural from an enumerative point of view; how many elements does $[3]$ have ? Well, $3$.

The notation $2^{[n]}$ to denote the set of subsets of $[n]$ is then quite natural from an enumerative point of view; how many elements does $2^{[3]}$ have ? Well, $2^3=8$.

And of course these $8$ sets are $ \phi , \{1 \},\{2 \},\{3 \},\{1,2 \},\{1,3 \},\{2,3 \},\{1,2,3 \}$.

Donald Splutterwit
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