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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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There is no function from a non-empty set into an empty set. Could you explain what your actual question is? Do you want to count empty sub-sequences of a given set? –  Phira Nov 19 '12 at 12:52
The case of empty set is very important and must be considered. That maybe explain why for example $0!=1$ –  Adi Dani Nov 19 '12 at 13:01

1 Answer 1

up vote 5 down vote accepted

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.

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