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The notion of sequence is basic notion in combinatorics and can be defined

1.Mapping from a finite set for example $I_m=\{0,1,2,...,m-1\}$ to an other set $X$ is finite sequence with terms in $X$ and is denote by $s=(x_o,x_1,...,x_{m-1}), x_i\in X$ 2.Mapping from a countable set for example $\mathbb N=\{0,1,2,...,n,...\}$ to an other set $X$ is infinite sequence and is denoted by $s=(x_0,x_1,...,x_n,...), x_i\in X$

Finite sequnces with terms from a finite set can be enumerated and they we call permutations or variations. Definition of sequences s contain two cases that are extremal.

$s:\emptyset\to X$ $s:X\to\emptyset$ $X$ is countable

How to deal with such cases. Any suggestions

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    The case of empty set is very important and must be considered. That maybe explain why for example $0!=1$2012-11-19

1 Answers 1

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There is exactly one map $s \colon \emptyset \to X$ for any set $X$, it is called the empty map. In the usual interpretation of maps as subsets (here of $\emptyset \times X = \emptyset$) it corresponds to the empty set. In terms of sequences, you can count it as the empty sequence (with zero terms) and denote it $s = ()$. It is for example important as neutral element of juxtraposition in the monoid of all finite sequences in $X$.

On the other side, if $X$ is not empty, then there is no map $X \to \emptyset$. Such a map, given by a set $A \subseteq X \times \emptyset = \emptyset$ must have $ \forall x \in X \; \exists! y\in \emptyset\; (x,y) \in A = \emptyset. $ As for non-empty $X$ this can't be true, there is no such map.