User:Xiong/math

(posted on )

Let S be any finite set and suppose:

$$ x \notin S $$

Let:

$$ K = S \cup \left\{ x \right\} $$

1. Prove that $$P(K)$$ is the disjoint union of $$P(S)$$ and $$ X = \left \{T \subseteq K : x \in T \right\} $$.

That is, show that $$ P(K) = P(S) \cup X$$ and $$P(S) \cap X = \emptyset $$

2. Prove that every element of $$X$$ is the union of a subset of $$S$$ with $$\{x\}$$, and that if you take different subsets of $$S$$ you get different elements of $$X$$.

Argue that, therefore, $$X$$ has the same number of elements as $$P(S)$$.

3. Argue that the previous two parts allow you to conclude that if $$S$$ is a finite set, then $$P(K)$$ has twice as many elements as $$P(S)$$.