Abstract
The basic notational conventions used in this book are described. Composition of mappings is defined the standard left-handed way: \(f\ g\) means mapping g was applied first. But things are a little more complicated than that since we must also deal with both left and right operators, binary operations and monoids. For example, right operators are sometimes indicated exponentially—that is by right superscripts (as in group conjugation)—or by right multiplication (as in right R-modules). Despite this, the “\(\circ \)”-notation for composition will always have its left-handed interpretation. Of course a basic discussion of sets, maps, and equivalence relations should be expected in a beginning chapter. Finally the basic arithmetic of the natural and cardinal numbers is set forth so that it can be used throughout the book without further development. (Proofs of the Schröder-Bernstein Theorem and the fact that \(\aleph _0\cdot \aleph _0 = \aleph _0\) appear in this discussion.) Clearly this chapter is only about everyone being on the same page at the start.
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- 1.
We are deliberately vague in talking about parts rather than “elements” for the sake of generality.
- 2.
There is a common misunderstanding of this word “abstract” that mathematicians seem condemned to suffer. To many, “abstract” seems to mean “having no relation to the world—no applications”. Unfortunately, this is the overwhelming view of politicians, pundits of Education, and even many University Administrators throughout the United States. One hears words like “Ivory Tower”, “Intellectuals on welfare”, etc. On the contrary, these people have it just backwards. A concept is “abstract” precisely because it has more than one application—not that it hasn’t any application. It is very important to realize that two things introduced in distant contexts are in fact the same structure and subject to the same abstract theorems.
- 3.
Of course the sets A and B might have entirely different descriptions, and yet possess the same collection of members.
- 4.
This is not just a matter of silly grammatical style. How many American Calculus books must students endure which assert that a “function” (for example from the set of real numbers to itself) is a “rule that assigns to each element of the domain set, a unique element of the “codomain” set? The “rules” referred to in that definition are presumably instructions in some language (for example in American English) and so these instructions are strings of symbols in some finite alphabet, syllabary, ideogramic system or secret code. The point is that such a set is at best only countably infinite whereas the collection of subsets R of \(A\times B\) may well be uncountably infinite. So there is a very good logical reason for viewing relations as subsets of a Cartesian product.
- 5.
Note that there is no grammatical room here for a “multivalued function”.
- 6.
Unlike the notion of “restriction”, this construction does not seem to enjoy a uniform name.
- 7.
Note that in the notation, the “\(\alpha \)” is ranging completely over I and so does not itself affect the collection being described; it is what logicians call a “bound” variable.
- 8.
A “one-to-one correspondence” is not to be confused with the weaker notion of a “one-to-one mapping” introduced on p. 9. The latter is just an injective mapping which may or may not be a bijection.
- 9.
It should be clear to the student that this partial ordering is on the collection of cardinal numbers. It is not a relation between the sets themselves.
References
Cohn PM (1977) Algebra, vol 2. Wiley, London
Devlin K (1997) The joy of slets. Springer, New York
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Shult, E., Surowski, D. (2015). Basics. In: Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-19734-0_1
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