Arrays

Why

We name lists of lists, and so on.

Definition

Let $s$ be a sequence of natural numbers: $s = (n_1, \dots , n_d)$. An array of size (or shape) $s$ is a function whose domain is the set

\[ I = \Set{(m_1, \dots , m_d)}{ 1 \leq m_1 \leq n_1, \dots , 1 \leq m_d \leq n_d}. \]

We call the set $I$ the set of indices of the array. We call the codomain of the function the set of values of the array. If $A$ is the set of values, we say that the array is in A. We call the length of $s$ (here denoted $d$) the dimension of the array.

Case $d = 1$

If the shape of the array has length one, then the array is no different from a sequence. In this case, the terminology for arrays coincides with that for sequences.

Case $d = 2$

If the shape of the array has length two, then the array can is a table with $n_1$ rows and $n_2$ columns.¹ We say that the array is two-dimensional. We denote the $n_1 \times n_2$ array with elements in the set $A$ by $A^{n_1 \times n_2}$.

Simplified indices

In the case that $a$ is a one-dimensional array, or sequence, we use the common terminology $a_i$ for the $i$th element of $a$ In the case that $a$ is a two dimensional array, we write $a_{ij}$ for $a_{(i,j)}$.

Compare with Matrices. ↩︎