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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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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
Halmos PR (2001) Naive set theory, New edn. Springer, New York, chapters 8 – 10
Hein JL (2002) Discrete structures, logic and computability, 2nd edn. Jones and Bartlett, Boston, chapter 2
Lipschutz S (1998) Set theory and related topics, Schaum’s outline series. McGraw Hill, New York, chapters 4 – 5
Velleman DJ (2006) How to prove it: a structured approach, 2nd edn. Cambridge University Press, New York, chapter 5
- Associating One Item with Another: Functions
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