Abstract
One of the most essential ideas in modern mathematics is the concept of a function. A function is a way of associating each element of a set A with exactly one element of another set B. A precise set-theoretic definition of a function in terms of ordered pairs is presented and the meaning of a “well-defined” function is discussed. Proof strategies are introduced that show students how to proof that a function is one-to-one and how to prove that a function is onto. Afterwards, we prove that if a function one-to-one and onto, then it has an inverse function which is also one-to-one and onto. The composition of two functions is also defined and discussed. Given a subset of the domain (co-domain) of a function, the image (inverse image) of this subset is also presented. The last section in this chapter covers Cantor’s early work on the “size of infinite sets.”
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Notes
- 1.
“Putting the horse before the cart” is an expression that is used when the order of certain facts or ideas have been reversed.
- 2.
There is only one way that such a positive rational number can be written in reduced form (see Definition 3.8.7 and Exercise 6 on page 141). Thus, f is well-defined.
- 3.
The sequence \({n}_{1},{n}_{2},{n}_{3},\ldots \) can be defined by recursion using the Well-Ordering Principle 4.1.1.
- 4.
g(i) is the unique element a ∈ A such that f(a) = n i , the i-th element in R. See the proof of Theorem 6.2.11.
- 5.
We are tacitly using the axiom of choice to obtain the choice set \(\{{f}_{i} : i \in \mathbb{N}\}\) for the family \(\{{F}_{i} : i \in \mathbb{N}\}\), where F i is the set of one-to-one functions \({f}_{i}: {A}_{i} \rightarrow \mathbb{N}\) for each \(i \in \mathbb{N}\) (see page 156).
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Cunningham, D.W. (2013). Functions. In: A Logical Introduction to Proof. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3631-7_6
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