Countable v. Uncountable: Prove that B* , the set of all finite strings over B =
ID: 3854190 • Letter: C
Question
Countable v. Uncountable:
Prove that B*, the set of all finite strings over B = {0; 1} is countable, that is, B* can be put in a one-to-one correspondence with the natural numbers N = {0, 1, 2,3,.....}. Hint: List the strings in B* in order of increasing length. Notice where the strings reset to all 0’s.
strings |lambda 0 1 00 01 10 11 000 001 010 011 100 101 110 111 0000 0001
binary |0 0 1 0 1 2 3 0 1 2 3 4 5 6 7 0 1
n N |0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Please explain me the solution
Explanation / Answer
In set theory, XY is the set of all functions from Y to X. As "2" can be defined as {0,1} (see natural number), 2S (i.e., {0,1}S) is the set of all functions from S to {0,1}. By identifying a function in 2S with the corresponding preimage of 1, we see that there is a bijection between 2S and P(S), where each function is the characteristic function of the subset in P(S) with which it is identified. Hence 2S and P(S) could be considered identical set-theoretically. (Thus there are two distinct notational motivations for denoting the power set by 2S: the fact that this function-representation of subsets makes it a special case of the XY notation and the property, mentioned above, that |2S| = 2|S|.)
This notion can be applied to the example above in which S = {x, y, z} to see the isomorphism with the binary numbers from 0 to 2n 1 with n being the number of elements in the set. In S, a "1" in the position corresponding to the location in the set indicates the presence of the element. So {x, y} = 110.
For the whole power set of S we get:
Relation to binomial theorem[edit]
The power set is closely related to the binomial theorem. The number of subsets with k elements in the power set of a set with n elements is given by the number of combinations, C(n, k), also called binomial coefficients.
For example, the power set of a set with three elements, has:
Algorithms[edit]
If S is a finite set, there is a recursive algorithm to calculate P(S).
Define the operation F (e, T) = {X {e} | X T}.
In English, return the set with the element e added to each set X in T.
In other words, the power set of the empty set is the set containing the empty set and the power set of any other set is all the subsets of the set containing some specific element and all the subsets of the set not containing that specific element.
Subsets of limited cardinality[edit]
The set of subsets of S of cardinality less than is denoted by P(S) or P< (S). Similarly, the set of non-empty subsets of S might be denoted by P 1(S).
Power object[edit]
A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective the idea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structure or algebra.
Now the power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard subalgebras behave analogously to subsets.
However, there are two important properties of subsets that do not carry over to subalgebras in general. First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice. Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set {0,1} = 2, there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way.
Certain classes of algebras enjoy both of these properties. The first property is more common, the case of having both is relatively rare. One class that does have both is that of multigraphs. Given two multigraphs G and H, a homomorphism h: G H consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set HG of homomorphisms from G to H can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore, the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G to the multigraph definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of G as the multigraph G, called the power object of G.
What is special about a multigraph as an algebra is that its operations are unary. A multigraph has two sorts of elements forming a set V of vertices and E of edges, and has two unary operations s,t: E V giving the source (start) and target (end) vertices of each edge. An algebra all of whose operations are unary is called a presheaf. Every class of presheaves contains a presheaf that plays the role for subalgebras that 2 plays for subsets. Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed) and has an object , called a subobject classifier. Although the term "power object" is sometimes used synonymously with exponential object YX, in topos theory Y is required to be .
Functors and quantifiers[edit]
In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint.[3]
See also[edit]
Notes[edit]