General Sets
A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S the power set of S, i.e., the set of all subsets of S (here written as P(S)), is larger than S itself. This proof proceeds as follows:
Let f be any function from S to P(S). It suffices to prove f cannot be surjective. That means that some member T of P(S), i.e., some subset of S, is not in the image of f. As a candidate consider the set:
For every s in S, either s is in T or not. If s is in T, then by definition of T, s is not in f(s), so T is not equal to f(s). On the other hand, if s is not in T, then by definition of T, s is in f(s), so again T is not equal to f(s). For a more complete account of this proof, see Cantor's theorem.
Read more about this topic: Cantor's Diagonal Argument
Famous quotes containing the words general and/or sets:
“At that,
his small size,
keen eyes,
serviceable beak
and general truculence
assure his survival—”
—William Carlos Williams (1883–1963)
“This is certainly not the place for a discourse about what festivals are for. Discussions on this theme were plentiful during that phase of preparation and on the whole were fruitless. My experience is that discussion is fruitless. What sets forth and demonstrates is the sight of events in action, is living through these events and understanding them.”
—Doris Lessing (b. 1919)