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Associating One Item with Another: Functions

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Sets, Logic and Maths for Computing

Part of the book series: Undergraduate Topics in Computer Science ((UTICS))

Abstract

Functions occur everywhere in mathematics and computer science. In this chapter, we introduce the basic concepts needed in order to work with them.

We begin with the intuitive idea of a function and its mathematical definition as a special kind of relation. We then see how the general concepts for relations that were studied in the previous chapter play out in this particular case (domain, range, restriction, image, closure, composition, inverse) and distinguish some important kinds of function (injective, surjective, bijective) with special behaviour.

These concepts permit us to link functions with counting, via the equinumerosity, comparison and surprisingly versatile pigeonhole principles. Finally, we identify some very simple kinds of function that appear over and again (identity, constant, projection and characteristic functions) and explain the deployment of functions to represent sequences and families.

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Selected Reading

  • The texts listed at the end of Chap. 2 on relations also have useful chapters on functions. In detail:

  • Bloch ED (2011) Proofs and fundamentals: a first course in abstract mathematics, 2nd edn. Springer, New York, chapter 4

    Google Scholar 

  • Halmos PR (2001) Naive set theory, New edn. Springer, New York, chapters 8 – 10

    Google Scholar 

  • Hein JL (2002) Discrete structures, logic and computability, 2nd edn. Jones and Bartlett, Boston, chapter 2

    Google Scholar 

  • Lipschutz S (1998) Set theory and related topics, Schaum’s outline series. McGraw Hill, New York, chapters 4 – 5

    Google Scholar 

  • Velleman DJ (2006) How to prove it: a structured approach, 2nd edn. Cambridge University Press, New York, chapter 5

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Philosophy, London School of Economics, Houghton Street, WC2A 2AE, London, UK

    David Makinson

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  1. David Makinson
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Correspondence to David Makinson .

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© 2012 Springer-Verlag London Limited

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Makinson, D. (2012). Associating One Item with Another: Functions. In: Sets, Logic and Maths for Computing. Undergraduate Topics in Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-2500-6_3

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  • DOI: https://doi.org/10.1007/978-1-4471-2500-6_3

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-2499-3

  • Online ISBN: 978-1-4471-2500-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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