Abstract
This chapter introduces the essential mathematics for computing and discusses fundamental concept such as sets, relations and functions. Sets are collections of well-defined objects; relations indicate relationships between members of two sets A and B; and functions are a special type of relation where there is exactly (or at most) one relationship for each element a ∈ A with an element in B.
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Notes
- 1.
We distinguish between total and partial functions. A total function f : A → B n : ℕ in A whereas a partial function may be undefined for one or more values in A.
- 2.
There are mathematical objects known as multi-sets or bags that allow duplication of elements. For example, a bag of marbles may contain three green marbles, two blue and one red marble.
- 3.
The British logician, John Venn, invented the Venn diagram. It provides a visual representation of a set and the various set theoretical operations. Their use is limited to the representation of two or three sets as they become cumbersome with a larger number of sets.
- 4.
The natural numbers, integers and rational numbers are countable sets (i.e. they may be put into a one-to-one correspondence with the natural numbers), whereas the real and complex numbers are uncountable sets.
- 5.
Cartesian product is named after René Descartes who was a famous seventeenth-century French mathematician and philosopher. He invented the Cartesian coordinates system that links geometry and algebra, and allows geometric shapes to be defined by algebraic equations.
- 6.
Permutations and combinations are discussed in Chap. 5 of O’Regan (2017).
- 7.
Parnas made important contributions to software engineering in the 1970s. He invented information hiding which is used in object-oriented design.
- 8.
We distinguish between total and partial functions. A total function is defined for all elements in the domain whereas a partial function may be undefined for one or more elements in the domain.
- 9.
Higher order functions are functions that take functions as arguments or return a function as a result. They are known as operators (or functionals) in mathematics, and one example is the derivative function dy/ dx that takes a function as an argument and returns a function as a result.
- 10.
Monads are used in functional programming to express input and output operations without introducing side effects. The Haskell functional programming language makes use of this feature.
- 11.
This is the most common algorithm used to perform type inference. Type inference is concerned with determining the type of the value derived from the eventual evaluation of an expression.
- 12.
Lisp is a multi-paradigm language rather than a functional programming language.
- 13.
Goldbach was an eighteenth-century German mathematician and Goldbach’s conjecture has been verified to be true for all integers n < 12 × 1017.
- 14.
Pierre de Fermat was a seventeenth-century French civil servant and amateur mathematician. He occasionally wrote to contemporary mathematicians announcing his latest theorem without providing the accompanying proof and inviting them to find the proof. The fact that he never revealed his proofs caused a lot of frustration among his contemporaries, and in his announcement of his famous last theorem, he stated that he had a wonderful proof that was too large to include in the margin. He corresponded with Pascal and they did some early work on the mathematical rules of games of chance and early probability theory. He also did some early work on the Calculus.
- 15.
The transition function may be undefined for a particular input symbol and state.
- 16.
It may be a total or a partial function.
- 17.
The four-colour theorem states that given any map it is possible to colour the regions of the map with no more than four colours such that no two adjacent regions have the same colour. This result was finally proved in the mid-1970s.
- 18.
Königsberg (now called Kaliningrad) was founded in the thirteenth century by Teutonic Knights and was one of the cities of the Hanseatic League. It was the historical capital of East Prussia (part of Germany), and it was annexed by Russia at the end of the Second World War. The famous German philosopher, Immanuel Kant, spent all his life in the city and is buried there.
- 19.
The Church–Turing thesis states that anything that is computable is computable by a Turing machine.
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O’Regan, G. (2018). Overview Mathematics in Computing. In: World of Computing. Springer, Cham. https://doi.org/10.1007/978-3-319-75844-2_4
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