Writing About Sets - Essential Dictionary I

Mathematical Writing - Vivaldi Franco 2014

Writing About Sets
Essential Dictionary I

The vocabulary on sets developed so far is sufficient for our purpose. We begin to use it in short phrases which define sets.

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Note that we haven’t used any symbols. The set in item 1 is $$\mathbb {C}^2$$. In item 2, among the infinitely many points of the unit circle, we consider those having rational co-ordinates. There is no difficulty in writing this set symbolically:

$$ \{(x,y)\in \mathbb {Q}^2\,:\,x^2+y^2=1\} $$

although its properties are not obvious from the definition. This set is non-empty (the points $$(0,\pm 1),\,(\pm 1,0)$$ belong to it), but is it infinite? This example illustrates the power of a verbal definition. Item 3, which defines a subset of $$\mathbb {N}$$, makes an even stronger point. This set must be extremely large, but can we even show that it is non-empty? In item 4, each line counts as a single element, rather than an infinite collection of points (otherwise our set of lines would be the whole plane). The symbolic definition of this set is awkward; to simplify it we’ll consider suitable representations of this set (Sect. 2.3.3).

It is possible to specify a type of set, without revealing its precise identity. In each of the following sets there is at least one unspecified quantity.

·  The set of fractions representing a rational number.

·  The set of divisors of an odd integer.

·  A proper infinite subset of the unit circle.

·  The cartesian product of two finite sets of complex numbers.

·  A finite set of consecutive integers.

Next we define sets in two ways, first with a combination of words and symbols, and then with words only. One should consider the relative merits of the two formulations.

·  Let $$X=\{3\}$$.

·  The set whose only element is the integer 3.

·  Let $$X=\{m\}$$, with $$m\in \mathbb {Z}$$.

·  A set whose only element is an integer.

·  Let $$m\in \mathbb {Z}$$, and let $$X$$ be a set such that $$m\in X$$.

·  A set which contains a given integer.

·  Let $$X$$ be a set such that $$X\cap \mathbb {Z}\not =\emptyset $$.

·  A set which contains at least one integer.

·  Let $$X$$ be a set such that $$\#(X\cap \mathbb {Z})=1$$.

·  A set which contains precisely one integer.

In the first two examples the combination of ’let’ and ’$$=$$’ replaces an assignment operator. An expression such as ’Let $$X\mathop {=}\limits ^{\scriptscriptstyle \nabla }\{3\}$$’ would be overloaded.

The distinction between definite and indefinite articles is essential, the former describing a unique object, the latter an unspecified representative of a class of objects. In the following phrases, a change in one article, highlighted in boldface, has resulted in a logical mistake.

BAD:

A proper infinite subset of a unit circle.

BAD:

A set whose only element is the integer 3.

BAD:

The set whose only element is an integer.

BAD:

The set which contains precisely one integer.

As a final exercise, we express some geometric facts using set terminology.

·  The intersection of a line and a conic section has at most two points.

·  The set of rational points in any open interval is infinite.

·  A cylinder is the cartesian product of a segment and a circle.

·  The complement of the unit circle consists of two disjoint components.

The reader should re-visit familiar mathematics and describe it in the language of sets.

Exercise 2.1

For each of the following topics:

$$ \text {prime numbers,}\; \text {fractions}, \text {complex numbers,} $$

(i) write five short sentences; (ii) ask five questions. The sentences should give a definition or state a fact; the questions should have mathematical significance, and preferably possess a certain degree of generality. [ $$\not \varepsilon $$ ] 4

Exercise 2.2

Define five interesting finite sets. [ $$\not \varepsilon $$ ]

Exercise 2.3

The following expressions define sets. Turn words into symbols, using standard or Zermelo definitions. (Represent geometrical objects, e.g., planar curves, by their cartesian equations.)

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Exercise 2.4

The following expressions define sets. Turn symbols into words. [ $$\not \varepsilon $$ ]

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