1. Introduction and Overview
1 (a) Excluding the row and column keys, the associative array has $m = 4$ rows and $n = 4$ columns, so (b) the number of entries $mn = 4 \cdot 4 = 16$. The array has $0$ empty entries and the number of filld entries is $mn = 4 \cdot 4 = 16$.
2 (a) Each entry has a associated vertex, so the number of vertices is $20$. (b) The vertices through which the line passes are $v_{09}, v_{11}, v_{02}, v_{16}, v_{06}$ and $v_{29}$.
3 (a) The array is called square because the number of rows ($m$) is equal to the number of columns ($n$). (b) The array is called symmetric because the entries above the main diagonal are a transpose of the entries below the main diagonal.
4 (a) The number of rows ($m$) in the associative array is $3$ and the number of columns ($n$) is $3$. (b) There are $3$ genre vertices and artist vertices in the graph. (c) The total number of entries is $mn = 3 \cdot 3 = 9$. (d) The number of empty entries are those entries that are left blank is $mn - (1 + 1 + 1 + 1)$ $=$ $9 - 4$ $=$ $5$, where $(1 + 1 + 1 + 1)$ $=$ $4$ is the number of filled entries. (d) The number of edges correlates to the number of filled entries, i.e. the number of edges is $4$.
5 Expression that illustrates (a) associativity and (c) distributivity of $\oplus$ and $\otimes$ among $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ is $\mathbf{A} \otimes (\mathbf{B} \oplus \mathbf{C})$ $=$ $(\mathbf{A} \otimes \mathbf{B})$ $\oplus$ $(\mathbf{A} \otimes \mathbf{B})$. Expression that illustrates (b) commutativity is $(\mathbf{A} \oplus,\otimes \mathbf{B})$ $\oplus,\otimes$ $\mathbf{C}$ $=$ $(AB)C$ $=$ $A(BC)$ $=$ $\mathbf{A} \oplus,\otimes (\mathbf{B}$ $\oplus,\otimes$ $\mathbf{C})$.
6 It is reasonable to assume that the need for storing data into tables happened as a application of writing. It is also reasonable to assume that the needs are more or less the same, i.e. to keep record of things, but due to invention of computer, machine based record keeping has becoma a major techniques for record keeping.
7 Examples of different perspectives or views into observing same data are given in the chapter. For example, the assosiative arrays along with their graphs are examples of this. Other example are tables of air temperatures and their visualisations as, say, as climate heat maps.
8 Main goal of data processing system is to enable the capability to store, manipulate, write and read data in superhuman way. Two advantages of mathematical modeling of data is that mathematics is often scale invariant from infinitesimal to the infinite. Also expressing data using mathematics allows applying a very plentiful set of mathematical methods to be applied on data and information processing.
9 The two main mathematical operations performed on data are addition and multiplication. These operations are well defined when processing bits and their implementations in different computational environments is very often very efficient. This means that computers handle summation and multiplication effortlessly. It could be said that computers are machines optimized for doing summations and multiplications in nanoseconds, i.e. in a one billionth of a second.
10 Defining summation and multiplication for associative arrays allows the application of a lot of well known theory to associative arrays. This allows a lot of research and applications.