Suppose that a coordinate grid of size m ×n is given by the list of all pairs (x, y) of integers

step 1 Let's write the matrix of the coefficients, big x, has the first column is all ones, and the sec

Suppose that a coordinate grid of size m ×n is given by the...

Suppose that a coordinate grid of size $m \times n$ is given by the list of all pairs $(x, y)$ of integers such that $0 \leqslant x \leqslant m$ and $0 \leqslant y \leqslant n$. Using a list comprehension, define a function grid $::$ Int $\rightarrow$ Int $\rightarrow$ [(Int, Int)] that returns a coordinate grid of a given size. For example: $$ \begin{aligned} & >\text { grid } 12 \\ & {[(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)]} \end{aligned} $$

Suppose that a coordinate grid of size $m \times n$ is given by the list of all pairs $(x, y)$ of integers such that $0 \leqslant x \leqslant m$ and $0 \leqslant y \leqslant n$. Using a list comprehension, define a function grid $::$ Int $\rightarrow$ Int $\rightarrow$ [(Int, Int)] that returns a coordinate grid of a given size. For example:
$$
\begin{aligned}
& >\text { grid } 12 \\
& {[(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)]}
\end{aligned}
$$

Key Concepts

List Comprehension

List comprehensions in functional programming, particularly in Haskell, provide a concise way to generate and transform lists. They allow the creation of new lists by specifying a source list or multiple source lists, applying a predicate for filtering, and an expression to generate elements. This method is highly expressive and clean, making it especially useful for problems where one needs to enumerate combinations or compute values based on extracted elements from existing sequences.

Cartesian Product

The Cartesian product is a fundamental concept in set theory and combinatorics that involves generating all possible ordered pairs from two sets. In the context of generating a coordinate grid, this concept is used to pair every element from one range (representing one dimension) with every element from another range (representing the other dimension). This method ensures that every coordinate within the specified limits is included in the grid.

Function Type Signature

In Haskell, the function type signature is an explicit declaration of the types of input parameters and the type of the output. This feature not only serves as documentation but also allows the compiler to enforce type correctness. For instance, declaring a function that takes two integers and returns a list of pairs (tuples) of integers provides clarity and safety by clearly outlining the expected types.

Tuple Construction

Tuples in Haskell are used to store fixed collections of elements, where each element can be of a different type. They are particularly useful for representing coordinates, such as (x, y) pairs, where both x and y are integers. This structure enables grouping of related values into a single compound value that can be easily manipulated or returned by functions.

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