Difference between revisions of "Power set"
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The power set forms a [[Boolean algebra]] with the operations of [[union of sets]], [[intersection of sets]] and [[relative complement]]. | The power set forms a [[Boolean algebra]] with the operations of [[union of sets]], [[intersection of sets]] and [[relative complement]]. | ||
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+ | If $f$ is a map from $X$ to $Y$ then there are associated maps $f_\vdash : \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ and $f^\dashv : \mathcal{P}(Y) \rightarrow \mathcal{P}(X)$ defined for $A \in \mathcal{P}(X)$, $B \in \mathcal{P}(Y)$ by | ||
+ | $$ | ||
+ | f_\vdash(A) = \{ y \in Y : \exists a \in A\,,\, y = f(a) \} \ ; | ||
+ | $$ | ||
+ | $$ | ||
+ | f^\dashv(B) = \{ x \in X : \exists b \in B\,,\, b = f(x) \} \ . | ||
+ | $$ | ||
+ | Alternative notations are $f[A]$, $f^{-1}[B]$ respectively. | ||
====References==== | ====References==== |
Revision as of 11:53, 20 March 2016
2020 Mathematics Subject Classification: Primary: 03E [MSN][ZBL]
of a set $X$
The set of all subsets of $X$, denoted $\mathcal{P}(X)$. One has $A \in \mathcal{P}(X) \Leftrightarrow A \subseteq X$. The power set of a finite set of $n$ elements has $2^n$ elements. Cantor's theorem states that a set and its power set can never be put into one-to-one correspondence, hence cannot have the same cardinality.
The power set forms a Boolean algebra with the operations of union of sets, intersection of sets and relative complement.
If $f$ is a map from $X$ to $Y$ then there are associated maps $f_\vdash : \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ and $f^\dashv : \mathcal{P}(Y) \rightarrow \mathcal{P}(X)$ defined for $A \in \mathcal{P}(X)$, $B \in \mathcal{P}(Y)$ by $$ f_\vdash(A) = \{ y \in Y : \exists a \in A\,,\, y = f(a) \} \ ; $$ $$ f^\dashv(B) = \{ x \in X : \exists b \in B\,,\, b = f(x) \} \ . $$ Alternative notations are $f[A]$, $f^{-1}[B]$ respectively.
References
[1] | P. R. Halmos, Naive Set Theory, Springer (1960) ISBN 0-387-90092-6 |
Power set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power_set&oldid=37939