Chapter 9  Scala and mathlib
Learning how to read and write code is quite similar to learning how to use math
to formalise verbal theories. In this chapter we cover the basics of coding in
Scala and mathlib
. Understanding Scala code might seem daunting at first,
but you have already mastered a more difficult skill: Translating verbal theory
(which is underspecified) to formal, mathematical characterizations. Translating
from formalizations to simulation code requires understanding Scala, but it is
easier in the sense that there is much less ambiguity to deal with.
At the end of this chapter you will be able to understand how concepts from Chapter 3  Math concepts and notation can be translated to Scala code. The chapter is quite detailed and we recommend to use it as a reference guide when reading chapters on simulations or when writing your own code. The cheat sheet down below provides a helpful overview.
Scala fundamentals
We cover only a small subset of the full Scala language here. This subset suffices for implementing basic simulations of formal theories. If after reading this chapter you want to learn more about Scala, there are various textbooks (e.g. Odersky, Spoon and Venners, 2019; but see also Scala books) and online resources such as the official Tour of Scala and Creative Scala.
Variables, functions and types
The basic expressions in a functional programming language such as Scala are
variables, functions and types. Variables, like their mathematical counterpart,
are containers that can store a value.Variable is not completely accurate terminology here. val
stores a value which is (in functional programming terminology) immutable, i.e., it cannot change. It is more like a constant, rather than a variable. In this book, we will rarely use mutable var
variables. For example:
val myNumber = 3
Let’s break this example down into its parts. The word val
tells the
computer that you will specify a value, which is a container storing a
constant (in this case the integer 3). The identifier (name) of this value is
the word myNumber
. The assignment operator =
takes the value on its
right side and stores is in the container on the left side.
Any container in Scala has a type. For example, the value myNumber
above has type Int
which stands for integer. Scala can often derive these
types automatically, but for clarity you may want to use explicit types:
val mySecondNumber: Int = 2
Types are helpful. They provide a safety net against programming mistakes
since you cannot assign a value that is incompatible with the specified type.
Running the following code results in a type mismatch
error and the
compiler (i.e., the software that interprets and runs your code) will even tell
you which type it expected and which type you provided.
val mySecondNumber: Int = 3.9
Functions allow us to write code that takes input, one or more arguments, and returns output. For example, addition \(add(x, y) = x+y\) can be coded as:
def add(x: Int, y: Int): Int = {
x + y
}
add(3, 4)
The word def
specifies that we are constructing a function with the
identifier add
. The commaseparated list between parantheses is the list
of arguments you can pass to this function, where each argument has a specified
type. Functions require that the type of the output is explicitly defined, in
this example Int
. Then the assignment operator =
links the body of
function which is delineated by curly brackets. Whenever we call this function
with the right arguments, the value of the body is computed relative to the
arguments and that value is the output of the function.
Fill in the blanks in the code below and write a function that computes the following equation: \(f(a, b, c) = a + b * c\)
def equation(...): Int = {
???
}
equation(2, 5, 1) == 3 // Test the function, true if correct.
Functions in Scala have types too which becomes clearer with the following alternative notation.
def add: (Int, Int) => Int = (x: Int, y: Int) => {
x + y
}
add(3, 4)
This notation is not used often since it is hard to read, but it explicitly defines the function’s type which is very similar to how we express function types mathematically:
Scala  (Int, Int) => Int 

Math  \(add: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}\) 
In functional programming languages like Scala you can even pass a function as an argument of another function. This is a powerful way to organize your code and very useful in writing simulation code that is closely tied to formal theories. In a sense, a theory is a function itself, mapping a list of arguments (input) to output. We’ve seen that some formal theories can have functions as arguments. For example, Selecting invitees (version 1) in Chapter 4 takes as input the function \(like: P \times P \rightarrow \{true, false\}\). Here is a (partial) example of a function as argument.
def selectingInvitees(..., like: (Person, Person) => Boolean)
We return to selecting invitees in the next chapter, where we will use this coding strategy. For now, we should note one syntactic oddity in the Scala language. Sometimes when you pass a function as an argument, the compiler will complain with the following message:
error: missing argument list for method myFun
Unapplied methods are only converted to functions when a function type is expected.
You can make this conversion explicit by writing `myFun _` or `myFun(_)` instead of `myFun`.
The solution for this problem is often following the instructions in the error
explicitly and add an underscore _
to the function. For example:
selectingInvitees(..., like) // Procudes error.
selectingInvitees(..., like _) // Correct.
Finally, while we won’t go into details of objectorientation here, it is useful
to know that some functions accompany certain types. For example, the type
String
has functions builtin that can be called with the dotnotation.
These functions (also called methods) have access to the value they are called
upon. The following example called method toUpperCase
that evaluates to
the uppercase version of the original string.
"This is a String.".toUpperCase
Autocomplete can be triggered with
Ctrlspace
or Cmdspace
in Scalafiddle which we use in this book.
Other development tools often use the same keycommand, or alternatively the
Tab
key. Many Scala development tools allow you to look through the
list of methods by using auto completion. Just add a .
to a value and
access auto complete to open the list.
Blocks and scope
We have implicitly used the notions of block and scope, but how are they defined? In Scala a block is a sequence of expressions *delineated by curly brackets. A block has a value which is the value of the *last statement in the block.
{
val a = 3
val b = 6
a + b // This block evaluates to 9 with type Int.
}
Blocks can be nested, but values and functions defined within a block cannot be accessed outside that block. This property is known as scope, accessing values or functions out of scope results in a compiler error:
{
val x = 3
{
val y = 6
x + y // Valid, x and y are in this block's scope.
}
x + y // Invalid, y is outside this block's scope.
}
Conditional
The conditional expression is more colloquially known as the ifthenelse
expression. It allows for branching paths of code, depending on the truth
value of the conditional. An example (try changing the value of x
):
val x = 4
if(x % 2 == 0) {
println("X is even.")
} else {
println("X is odd.")
}
Each part of the conditional consists of a code block, though for single expressions the curly brackets to simplify the code:
val x = 4
if(x % 2 == 0) println("X is even.")
else println("X is odd.")
You can have an arbitrary number of branching paths using else if
:
val x = 4
if(x < 0) println("X is negative.")
else if(x == 0) println("X is zero.")
else if(x <= 10) println("X is small.")
else println("X is large.")
The conditional expression is a block with nested blocks. This means that it
evaluates to the value of the last expression in the block that is evaluated
by the conditional. This behaviour is useful when defining a function whose
output is computed differently depending on some truth condition. For example,
a function that computes the absolute value of x
multiplies x
with
1 if x
is negative and otherwise evaluates to x
.
Complete the code below by implementing the body of abs(x: Int): Int
. The function should evaluate to the absolute (positive) value of
x
.
def abs(x: Int): Int = {
???
}
abs(10) == 10 // Test the function, true if correct.
Basic types
Scala comes with a plethora of types and datastructures, many of which fall beyond the scope of this book. However, the following basic types and their operators will be very useful to know.
Type  Math equivalent  Example value 

Int 
\(\mathbb{N}\)  3 
Double 
\(\mathbb{R}\)  2.7 
Boolean 
\(\{true, false\}\)  true 
Char 
n.a.  ‘c’ 
String 
n.a.  “Awesome” 
Int
and Double
share many operators such as addition +
,
subtraction 
, multiplication *
and division /
. The library
Math
also contains several useful functions which you can apply with the
dot notation:
val x: Double = 3
val y: Double = 6
println(x + y)
println(x  y)
println(x * 3)
println(y / 2) // Division.
println(y % 2) // Remainder or modulo.
println(Math.pow(x, y)) // Exponentiation, x^y.
println(Math.min(x, y))
println(Math.max(x, y))
For Boolean types these are some common expressions:
println(true && true) // Logical and.
println(true  false) // Logical or.
println(true ^ true) // Logical xor.
println(!true) // Negation.
And these can be combined with numbers like so:
val x: Double = 3
val y: Double = 6
println(x <= 3 && x > 1) // Number between 1 and 3 (inclusive).
println(x % 2 == 0  x > 0) // Positive even number.
println(x % 2 == 0 ^ y % 2 == 0) // Either x or y is even, not both.
println(!(x < 0)) // x is positive.
If you change in the code example above Double
to Int
you might
notice that there is a certain compatility between the two types. That is, the
compiler does not give a type error when you add a double to an integer, even
though the addition function expects two doubles or two integers. This property is known as polymorphism and is an
advanced topic. For the curious, see this overview on
Wikipedia. This is because Scala knows it can convert an integer to a double value, which
it will automatically do.
val x: Int = 3
val y: Double = 2.5
x+y // Evaluates to a Double value 5.5
Some types cannot be converted and when you try to mix these, the compiler will let you know with an error:
val x: Int = 3
val b: Boolean = true
x + b // How to add a Boolean to an Int?
This conversion also kicks in when calling a function with compatible arguments:
def add(a: Double, b: Double): Double = a +b
add(1.4, 3)
This conversion often only works one way. It is not possible to convert a Double
to an Int without loosing information. Observe what happens when you run the
code example above after changing the type of the arguments of add
to
Int
.
Lists, sets and tuples
When you want to store multiple values you can use collections. Example collections could be the temperature forecast for the next seven days:
22.2 °C 
23.1 °C 
23.7 °C 
22.3 °C 
24.3 °C 
24.7 °C 
25.1 °C 
Or the people that you know: Erik, Lamar, Angelica, Emanuel, Lorraine, Meghan, Myron, Erica, Lester, Javier, Kelly, Abraham, Lindsay, Harriet, and Guadalupe. Or the cost of a menu card item: Vegan pie costs €9,90.
Some collections, like the temperature forecast and menu card item, are ordered: one value follows the next. In math, these are expressed in a list or sequence \(\langle 22.2, 23.1, 23.7, 22.3, 24.3, 24.7, 25.1 \rangle\) or a tuple \((\text{vegan pie}, 9.9)\). Unordered collections, such as people, are expressed in a set \(\{\text{Erik},\) \(\text{Lamar},\) \(\text{Angelica},\) \(\text{Emanuel},\) \(\text{Lorraine},\) \(\text{Meghan},\) \(\text{Myron},\) \(\text{Erica},\) \(\text{Lester},\) \(\text{Javier},\) \(\text{Kelly},\) \(\text{Abraham},\) \(\text{Lindsay},\) \(\text{Harriet},\) \(\text{Guadalupe}\}\).
In Scala we can store ordered collections in a List
or tuple and unordered
collections in a Set
:
val forecast = List(22.2, 23.1, 23.7, 22.3, 24.3, 24.7, 25.1)
val menuItem = ("Vegan pie", 9.90)
val people = Set("Erik", "Lamar", "Angelica", "Emanuel", "Lorraine",
"Meghan", "Myron", "Erica", "Lester", "Javier", "Kelly",
"Abraham", "Lindsay", "Harriet", "Guadalupe")
We’ll dive into sets below using the mathlib
library, so let’s first get
some familiarity with lists. Some basic examples are in the code block below.
Try playing around with them to see what they do.
val forecastThisWeek = List(22.2, 23.1, 23.7, 22.3, 24.3, 24.7, 25.1)
val forecastNextWeek = List(22.3, 19.8, 18.4, 18.0, 17.6, 17.5, 17.2)
println(23.1 :: forecastThisWeek) // Prepend element.
println(forecastThisWeek ::: forecastNextWeek) // Prepend list.
println(forecastThisWeek.size) // Number of elements in list.
println(forecastThisWeek.contains(23.7)) // Does the list contain element?
println(forecastThisWeek.head) // The first element of the list.
println(forecastThisWeek.tail) // Everything except the first element.
println(forecastThisWeek(3)) // The nth element.
println(forecastThisWeek.isEmpty) // Checks whether the list is emtpy.
The power of collections lies in being able to apply to all of the elements. The
idea is that if we have a function that applies to one element, e.g., \(sq(x) =
x^2)\), we can apply it to all elements in the list. The most common of these
applications is called map
. It takes as argument a function f
and
evaluates to a list where each element computed using f
:
list 
1 
2 
3 
4 
\(\downarrow\)  \(\downarrow\)  \(\downarrow\)  \(\downarrow\)  
sq(1) 
sq(2) 
sq(3) 
sq(4) 

\(\downarrow\)  \(\downarrow\)  \(\downarrow\)  \(\downarrow\)  
list.map(sq) 
1 
4 
9 
16 
// Function that computes the square root of x.
def sq(x: Int): Int = x * x
val list = List(1, 2, 3, 4)
println(list.map(sq))
The type of the argument of the function must be the same type as the elements
in the list, but its output can be of any type. For example, take an Int
and return a String
.
// Function that creates a String with x "x"s.
def xx(x: Int): String = {
if(x==1) "x"
else "x" + xx(x1)
}
val list = List(1, 2, 3, 4)
println(list.map(xx))
What other useful things can we do with lists? Below are some selfexplanatory examples.
// Function that checks whether x is even.
def isEven(x: Int): Boolean = {
x % 2 == 0
}
val list = List(1, 2, 3, 4)
println(list.exists(isEven)) // Does list contain an element that isEven?
println(list.exists(_ > 3)) // Implicit function, does the list contain an element larger than 3?
println(list.forall(isEven)) // Do all elements in list return true for isEven?
println(list.forall(_ <= 100)) // Implicit function, does the list contain an element larger than 3?
println(list.filter(isEven)) // Filter out all elements that return true for isEven.
println(list.filter(_ < 3)) // Filter out all elements less than 3.
There are some special methods for lists that contain numbers:
// Function that checks whether x is even.
val list = List(1, 2, 3, 4)
println(list.sum) // The sum of all numbers in list.
println(list.product) // The product of all numbers in list.
Many of the functions and methods that work for lists, also work for other
collections such as sets. Even a String
can be treated as a
collection as it is essentially a list of characters.
Look at the code scaffold below. Implement the body of the function
consonant
and choose the method to apply to sentence
such that the code evaluates to the sentence with only consonants.
val sentence = "Hi, you're doing awesome!"
val vowels = List('a', 'e', 'i', 'o', 'u')
def consonant(character: Char): Boolean = {
???
}
sentence.___(consonant)
Generics
Collections in Scala are what is called generic. The change their type
depending on the values you apply to them. The list containing the forecast
consists of values of type Int
, therefore the type of forecast
is
List[Int]
. In Scala, the type of a generic includes the type of its
contents denoted between square brackets.
What benefits do we get from generics?
Collections, courtesy of being generic, can store different types. A set of
doubles? No problem Set[Double]
. A list of Booleans? Easy
Set[Boolean]
! The type of a collection also grants a level of protection
to the programmer. You cannot prepend a Boolean to a list of integers:
val forecast: List[Double] = List(22.2, 23.1, 23.7, 22.3, 24.3, 24.7, 25.1)
val forecast2: List[Double] = true :: forecast
If you could mix different (incompatible) types in a collection, performing
calculations on that collection would be either undefined (e.g., what does
mean to compute the sum of a list of Int
, Boolean
and String
?)
or it would have unpredictable effects.
mathlib
mathlib
is a library written to support the development of simulations of
formal theories (computationallevel models specifically). Admittedly, an exact implementation is not always
possible. Especially when the simulation deviates from the formal theory, it is
important to be able for experts to understand how the simulation is different. Ideally, we want (experts) to be able to read simulation code and
understand that it is an exact implementation of the formal theory. As you may
have noticed during the Scala introduction, functional programming is closely
related to the mathematical language of formal theories. Consider the following
example (toy) formal theory:
Pizza Toppings
Input: A set of toppings \(T\), a budget \(b\in\mathbb{N}\), and a cost function
for toppings \(c: T\rightarrow \mathbb{N}\)
Output:
A selection of toppings \(T'\subseteq T\) such that \(\sum_{t \in T}c(t) \leq b\).
In this section you will learn how to read (and write) simulation code that
implements a formal theory and is easy to understand that it does what we say it
does. Take a look at the following code implementing Pizza Toppings
. (We assume toppings are represented by Strings
.)
def pizzaToppings(toppings: Set[String], budget: Int, cost: String => Int): Set[String] = {
// Helper function to compute the cost of a subset of toppings.
def subsetCost(subset: Set[String]): Int = {
subset.map(cost) // Transform each element in subset to its cost using the cost function.
.sum // Sum all costs.
}
// Helper function to check if a given subset fits within the budget.
def subsetWithinBudget(subset: Set[String]): Boolean = {
subsetCost(subset) <= budget
}
powerset(toppings) // The set of all possible subsets.
.filter(subsetWithinBudget) // Filter (keep) all subsets that fit the budget.
.random // Get a random subset.
.getOrElse(Set.empty) // If to random subset exists, return the emtpy set.
}
This code example combines several expressions we covered earlier, but by breaking down the components it is relatively easy to show that the code implements Pizza Toppings . Of course, even though math is much less ambiguous as verbal theory, that does not exclude there being multiple (equivalent) expressions in math. Some might be easier to translate to Scala code than others. We will encounter examples where rewriting the mathematical expressions of the formal theory can help in clarifying the relationship with the simulation.
In the next section, we explore how mathlib
allows writing code for formal
theories using set theory. The library also contains support for probability and
graph theory. You can find out more at the library Github
page, but using these advanced
mathematics requires installing a local development environment such as
Jupyter/Almond or Intellij (see Installing Scala and mathlib
).
Set theory
In this book, set theory plays an important role in formalizing verbal theories,
so we start exploring mathlib
there. This section follows the same
structure as Chapter 3  Math concepts and notation.
Creating a set is similar to creating other collections such as lists. Set
membership in Scala is an expression that evaluates to true
if the element
is within the set or false
if it is not. The Scala code below creates the
set \(P = \{\text{Ramiro},\text{Brenda},\text{Molly}\}\), then tests if
\(\text{Ramiro}\in P\) and if \(\text{Saki}\in P\).
val p: Set[String] = Set("Ramiro", "Brenda", "Molly")
println("Ramiro" in p)
println("Saki" in p)
Subset and superset expressions also evaluate to true
or false
.
val animals = Set("cat", "cuttlefish", "turtle", "blue whale")
val mammals = Set("cat", "blue whale")
val thingsOnEarth = Set("cat", "cuttlefish", "turtle", "blue whale", "university", "chair")
println(mammals < animals) // Mammals is a subset of animals.
println(thingsOnEarth > animals) // Things on Earth is a superset of animals.
Intersection, union and difference evaluate to the correct super or subset.
val yourFriends = Set("John", "Roberto", "Holly", "Doris", "Charlene")
val myFriends = Set("Vicky", "Charlene", "Ramiro", "Johnnie", "Roberto")
println(yourFriends /\ myFriends) // Common friends using intersection.
println(yourFriends \/ myFriends) // Friends we know together using union.
println(myFriends \ yourFriends) // Who I know that you don't, using difference.
Before we look at set builder notation, consider the following. In formal theories, we often use set theoretic notation not as Boolean tests, but to stipulate the output. E.g., in Pizza Toppings , the output is a subset \(T'\subseteq T\) that fits in budget. This notation, however, does not translate directly into Scala code. Consider the following rewrite of the formal theory output:
\(T' \in \mathcal{P}(T)\) such that \(\sum_{t\in T'}c(t)\leq b\)
This notation still does not easily translate, for two reasons. Firstly, because the relationship between \(T'\) and the desired property (fits in budget) is expressed in natural language. Secondly, because there are multiple outputs possible. Since the theory does not explicitly state which from the possible outputs is preferred, we need to assume it either returns the entire set or selects at random.
We can address the first issues by expressing the set of possible outputs and the relationship between \(T'\) and budgetfit explicitly in math using set building notation. The practice of decomposing parts of a theory (or code) into subparts is well known in mathematics and programming. It can help make math and code more readable, because we can abstract from the inner workings of a function. We make our lives easier by defining a help function for cost of subsets: \(cost: \mathcal{P}(T) \rightarrow \mathbb{N}\), where \(cost(T')=\sum_{t\in T'}c(t)\). Then the set of all subsets within budget can be defined as:
\[\left\{T'~\middle~\mathcal{P}(T) \wedge cost(T')\leq b\right\}\]Here, \(\mathcal{P}\) is the powerset notation. It denotes the set consisting of all possible subsets and \(T'\) is an element in that set. The setbuilder notation describes the set of all possible subsets that satisfy the budget constraint. Dealing second issue (multiple possible outputs), we make the decision to return a random valid output.
def pizzaToppings(toppings: Set[String], budget: Int, cost: String => Int): Set[String] = {
The helper function that computes the cost of a subset is implemented as:
// Helper function to compute the cost of a subset of toppings.
def subsetCost(subset: Set[String]): Int = {
sum(subset, cost)
}
We add one more helper function to check if a subset fits within the budget.
// Helper function to check if a given subset fits within the budget.
def subsetWithinBudget(subset: Set[String]): Boolean = {
subsetCost(subset) <= budget
}
Finally, we can define the output of Pizza Toppings using set builder notation:
{ powerset(toppings)  subsetWithinBudget _ }.random.getOrElse(Set.empty)
}
You may have noticed this code is different from the example given at the start of this section. Similarly to multiple mathematical expressions being equivalent, there exist (many) different ways of implementing the same function. This is a example implementation uses set builder to construct a set of subsets that fit within budget. Set builder, in essence, filters out all elements (in the case above the elements are sets) from the given set that do not satisfy the evaluation function on the right side.
{ givenSet  evalFun _ }
Complete the code scaffold below such that the set builder expression evaluates
to the subset of odd numbers in numbers
.
// Function that checks if a number is odd.
def isOdd(n: Int): Boolean = {
???
}
val numbers: Set[Int] = (0 to 11).toSet // Set consisting of numbers 0 to 11
{ ___  ___ }
The final expression for working with sets is the cardinal product \(\times\). The cardinal product between two sets \(A\) and \(B\) returns a set with all possible pairs of elements in \(A\) and \(B\). Pairs, in Scala, are represented as tuples.
val pair: (Int, String) = (4, "Book")
The type of a tuple combines the types of the elements. The cardinal product between two sets yields a set with the subtype of the tuple.
val numbers: Set[Int] = Set(1, 2, 3, 4)
val items: Set[String] = Set("Book", "Candle", "Wine")
val itemNumberPairs: Set[(String, Int)] = items x numbers
println(itemNumberPairs)
Simulation architecture
Now that we’ve covered the basics for Scala and mathlib
we can go cover
what constitutes a computer simulation of a formal theory. To run a computer
simulation you minimally need two components: The implementation of the formal
theory and sample input (Guest & Martin, 2020). The advantage of simulations
are that you can let the computer do the hard work computing many inputoutput
mappings, but it is still a lot of work if have to supply each input manually.
Manually supplying input can still be useful for checking for errors in the code, or to investigate particular limit cases of the theory and initial explorations. We will use manual inputs throughout the next chapters.
However, a more complete simulation covers many inputs, preferably exploring a wide range of different cases. For example, with the Pizza Toppings example one might want to explore inputs ranging from few to many toppings, with many different cost functions and budget. In these cases it is useful to write a helper function that can generate input with various properties.
Unfortunately, like with formalizing verbal theory, there is not a single recipe for writing input generators. Generators are highly contingent on the theory and the parts of the input domain you wish to simulate. Within the context of this book, you don’t need to write your own generators, but over the next chapters you will use example input generators provided by us.
We can put most generators in one of two categories: unconstrained input generation or constrained input generation. Understanding the difference between the to types of generators is important, as it impacts the kinds of inputs one can generate, which in turn will impact the computer simulation results.
Any procedure for randomly generating input will likely be biased. That means that some inputs are more likely to be generated than others. This can be due to particular structure in the input space, or due to properties of the generator.
A simplified representation of this bias is shown in the figure below. Out of the set of all possible inputs, the darker areas might be more likely to be generated than the light. Of course, input bias will also bias the output of the simulations.
We can deal with this bias by analyzing how well the generated inputs cover the parts of the space you are interested in. An unconstrained input generator is best used as such. These generators have bias over which we have no control such as in the figure above. Bias doesn’t render the simulations useless, as long as we can characterize the bias and have some understanding of how it impacts the results and limits our conclusions.
Alternatively, we can try to implement a constrained input generator. These types of generators are still biased, but we have some degree of control over the inputs they generate. Since there is still bias, it is wise to take that into account, but we may be better able to cover the input space by setting the right constraints. We can even use multiple sets of constraints to cover more area of the input space, as is shown in the figure below.
So why don’t we always write constrained input generators? This is because they are hard to make, at least harder than the unconstrained versions. It is a question we need to ask when developing simulations: Does the bias in the unconstrained generator skew the results too much? Then consider investing in developing a constrained input generator. For now, you can lean back and enjoy the generators we provide.
Cheat sheet
/**** BASIC TYPES ****/
val integer: Int = 3
val double: Double = 2.7
val bool: Boolean = true
val character: Char = 'c'
val string: String = "Awesome"
val x: Int = integer
val y: Double = double
println(x + y)
println(x  y)
println(x * 3)
println(y / 2) // Division.
println(y % 2) // Remainder or modulo.
println(Math.pow(x, y)) // Exponentiation, x^y.
println(Math.min(x, y))
println(Math.max(x, y))
println(true && true) // Logical and.
println(true  false) // Logical or.
println(true ^ true) // Logical xor.
println(!true) // Negation.
/**** Lists ****/
val forecastThisWeek = List(22.2, 23.1, 23.7, 22.3, 24.3, 24.7, 25.1)
val forecastNextWeek = List(22.3, 19.8, 18.4, 18.0, 17.6, 17.5, 17.2)
println(23.1 :: forecastThisWeek) // Prepend element.
println(forecastThisWeek ::: forecastNextWeek) // Prepend list.
println(forecastThisWeek.size) // Number of elements in list.
println(forecastThisWeek.contains(23.7)) // Does the list contain element?
println(forecastThisWeek.head) // The first element of the list.
println(forecastThisWeek.tail) // Everything except the first element.
println(forecastThisWeek(3)) // The nth element.
println(forecastThisWeek.isEmpty) // Checks whether the list is emtpy.
val list = List(1, 2, 3, 4)
def sq(x: Int): Int = x * x // Function that computes the square root of x.
println(list.map(sq)) // Map each value in list to sq(_)
def isEven(x: Int): Boolean = x % 2 == 0 // Function that checks whether x is even.
println(list.exists(isEven)) // Does list contain an element that isEven?
println(list.exists(_ > 3)) // Implicit function, does the list contain an element larger than 3?
println(list.forall(isEven)) // Do all elements in list return true for isEven?
println(list.forall(_ <= 100)) // Implicit function, does the list contain an element larger than 3?
println(list.filter(isEven)) // Filter out all elements that return true for isEven.
println(list.filter(_ < 3)) // Filter out all elements less than 3.
println(list.sum) // The sum of all numbers in list.
println(list.product) // The product of all numbers in list.
/**** Tuples ****/
val menuItem: (String, Double) = ("Vegan pie", 9.90)
println(menuItem._1) // Get the first value of the tuple.
println(menuItem._2) // Get the second value of the tuple.
/**** Sets ****/
val animals = Set("cat", "cuttlefish", "turtle", "blue whale")
val mammals = Set("cat", "blue whale")
val thingsOnEarth = Set("cat", "cuttlefish", "turtle", "blue whale", "university", "chair")
println("cat" in animals) // Test if element is in the set.
println(mammals < animals) // Mammals is a subset of animals.
println(thingsOnEarth > animals) // Things on Earth is a superset of animals.
val yourFriends = Set("John", "Roberto", "Holly", "Doris", "Charlene")
val myFriends = Set("Vicky", "Charlene", "Ramiro", "Johnnie", "Roberto")
println(yourFriends /\ myFriends) // Common friends using intersection.
println(yourFriends \/ myFriends) // Friends we know together using union.
println(myFriends \ yourFriends) // Who I know that you don't, using difference.
println({ Set(0, 1, 2, 3, 4, 5)  isEven _ }) // Build a set of even numbers.
println(myFriends x animals) // Set of all pairs of friends and animals using cardinal product.
References
Guest, O., & Martin, A. E. (2021). How computational modeling can force theory building in psychological science. Perspectives on Psychological Science 16(4),789802.
Odersky, M., Spoon, L., & Venners, B. (2019). Programming in Scala, 4th Edition. Artima.