Difference between revisions of "Power set"
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| − | Note that this proof does not rely upon either the [[ | + | Note that this proof does not rely upon either the [[Continuum Hypothesis]] or the [[Axiom of Choice]]. It is a good example of a [[diagonal argument]], a method pioneered by the mathematician [[Georg Cantor]]. |
==See Also== | ==See Also== | ||
Revision as of 12:14, 7 September 2006
The power set of a given set 
is the set 
of all subsets of that set.
Contents
Examples
The empty set has only one subset, itself. Thus 
.
A set 
with a single element has two subsets, the empty set and the entire set. Thus 
.
A set 
with two elements has four subsets, and 
.
Similarly, for any finite set with 
elements, the power set has 
elements.
Size comparison
Note that for any nonnegative integer , 
and so for any finite set , 
(where absolute value signs here denote the cardinality of a set). The analogous result is also true for infinite sets (and thus for all sets): for any set , the cardinality 
of the power set is strictly larger than the cardinality 
of the set itself.
Proof
There is a natural injection 
taking 
, so 
.
Suppose for the sake of contradiction that 
. Then there is a bijection 
. Let 
be defined by 
. Then 
and since 
is a bijection, 
.
Now, note that 
by definition if and only if 
, so if and only if 
. This is a clear contradiction. Thus the bijection cannot really exist and 
so 
, as desired.
Note that this proof does not rely upon either the Continuum Hypothesis or the Axiom of Choice. It is a good example of a diagonal argument, a method pioneered by the mathematician Georg Cantor.
See Also
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