Abstract
This chapter provides an introduction to fundamental building blocks in mathematics 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 ϵ A with an element in B. A set is a collection of well-defined objects that contains no duplicates. A binary relation R (A, B) where A and B are sets is a subset of the Cartesian product (A × B) of A and B. A total function f: A → B is a special relation such that for each element a ϵ A there is exactly one element b ϵ B. This is written as f(a) = b. A partial function differs from a total function in that the function may be undefined for one or more values of A.
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Notes
- 1.
We distinguish between total and partial functions. A total function f :A → B is defined for every element 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, whereas the real and complex numbers are uncountable sets.
- 5.
Cartesian product is named after René Descartes who was a famous seventeenth 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.
We discuss permutations and combinations in Chap. 5.
- 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.
Iverson received the Turing Award in 1979 for his contributions to programming language and mathematical notation. The title of his Turing Award paper was “Notation as a tool of thought”.
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O’Regan, G. (2017). Sets, Relations and Functions. In: Concise Guide to Formal Methods. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-64021-1_4
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