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Direct Products
Set Numbers
Family Unions and Intersections
Needed by:
Decision Processes
Index Lists
Integer Partitions
Joint Probability Distributions
Maximum Likelihood Distributions
Natural Number Notation
Natural Summation
Number Factorizations
Number Partitions
Numbered Partitions
Ordered Undirected Graphs
Real Plane
Size of Direct Products
Typed Graphs
Undirected Paths
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We want to talk about several objects in order.


Suppose $A$ is a set. A list (or finite sequence, $n$-tuple, string, dataset) in $A$ (or of elements from or of $A$) is a function

\[ a: \set{1, \dots , n} \to A. \]

In other words, a list is a family whose index set is $\set{1, \dots , n}$. The length (or size) of the list is the size of its domain. The $k$th entry (or term, record) of $A$ is the result $a_k$ of $k$; here $k \in \set{1, \dots , n}$.


Since the natural numbers are naturally ordered, we denote lists using this order, from left to right, between parentheses. For example, we denote the function $a: \set{1, \dots , 4} \to A$ by $(a_1, a_2, a_3, a_4)$.

Orderings and numberings

Let $A$ be a set with $\num{A} = n$. We call a list $a: \set{1, \dots , n} \to A$ an ordering of $A$ if $a$ is invertible. In this case, we call the inverse of the ordering a numbering (or enumeration) of $A$. An ordering associates with each number a unique object and a numbering associates with each object a unique number (the object’s index).

Relation to Direct Products

A natural direct product is a product of a list of sets. We denote the direct product of a list of sets $A_1, \dots , A_n$ by $\prod_{i = 1}^{n} A_i$. If each $A_i$ is the same set $A$, then we denote the product $\prod_{i = 1}^{n} A_i$ by $A^n$. The direct product $A^n$ is the set of lists in $A$ .

Natural unions and intersections

We denote the family union of the list of sets $A_1, \dots , A_n$ by $\cup_{i = 1}^{n} A_i$. Similarly, we denote the intersection by $\cap _{i = 1}^{n} A_i$.


An index range for a list $s$ of length $n$ is a pair $(i, j)$ for which $1 \leq i < j \leq n$. The slice corresponding to $(i,j)$ is the length $j-i$ list $s'$ defined by $s'_1 = s_{i}$, $s'_2 = s_{i+1}, \dots, s'_{j} = s_{i + j-1}$.

We denote the $(i,j)$-slice of $s$ by $s_{i:j}$. If $i = 1$ we use $s_{:j}$ and if $j = n$ we use $s_{i:}$ as shorthands for the slices $s_{1:j}$ and $s_{i:n}$.

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