Chapter 2

Introduction to the

Rudimentary Concepts of Sets,

Syllogistic Logic,

and Quantification

§ 2.1 BASIC NOTATION AND CONCEPTS FOR SETS.

The theory of sets was developed by many different mathematicians, but reached a rigorous level by the nineteenth and early twentieth centuries through the work of Boole, Cantor, Zermelo, Fraenkel, Dedekind, Frege, Zorn, von Neumann, etc. It is the one of the basic building blocks and a foundation of higher level mathematics and gives the mathematician the power to communicate abstract ideas and thoughts succinctly, clearly, and in an organised manner. There are many different approaches to an introduction to sets. One approach is to intuitively introduce the subject; another is to rigorous introduce the subject axiomatically. We shall discuss the subject using a bit of both manners of introduction.

A set and an element are undefined concepts much that same way as point, line, and plane are in Euclidean geometry. However, we can intuit an understanding of a set by thinking about a well defined group or collection of well defined objects. Any object in the set is called an element of the set and is said to be a member of the set or the element belongs to the set. We say that a set consists of it elements or a set contains its elements. So, in terms of a hierarchy, we should consider the objects as secondary to the set - - the set is the grouping and the elements the individuals much as matter is made up of elements. However, for a set to be well defined must exist within a realm that has been specified. That specification of all possible elements to be discussed is called the universe (or domain of discourse). The universe is also an undefined concept in so far as we axiomatically allow for its existence (e.g.: a premise of our discussion of sets is always that a well defined universe has been specified). No discussion of sets can properly take place before the universe has been defined. If one discussed a set without specifying what the universe is, then ambiguity can enter into the discussion and the theory collapses.

Suppose a well defined universe has been defined.[1] Examples of sets are the set of all Morehouse students enrolled in 1996; the set of all real numbers greater than or equal to 5 but less than 8; the set consisting of the multiplicative identity (that is to say the set consisting of 1); the set of protons in a carbon atom; the set of rational numbers; the set of all U. S. presidents who served in the twentieth century; and, the set of pens produced by Parker pens on the fifth of January 2002 A.D.. Examples of aggregates that are not sets since there is ambiguity, inconsistency, opinion, subjectivity, or such nebulous understanding of the concept that rigor cannot be achieved include the “set” of all my hopes and dreams, the “set” of all good presidents who served in the twentieth century; the “set” of all numbers; the “set” of all sets; and, the “set” of all Morehouse men.

An example of the ambiguity that arises with the concept of set is when one forgets to specify a universe. Thus, when described, the “set” is not well defined because a full accounting of elements has not preceded the discussion. Let us say we are in an arithmetic class and the instructor asks us to describe the set of numbers between 1 and 4. Many children would say, “2 and 3.” Others might say, “2, 1.5, 2.5, and so forth.” Others might express the numbers as fractions. Very few would consider that , p, e, etc. could be in the ‘set.’ The ambiguity arises because the universe has not been specified and the term “number” is not a singular concept in this context (indeed, how many children would realise that is a number (albeit complex))? So we can see that specification of a universe is important. Note that the universe, since it is a collection of elements that will be discussed, is a set. It is in our context the “biggest” set (big, large, etc. are very dangerous words when applied to sets - -be careful the term is not well defined; but, it should be intuitively appealing and can do no harm so long as one realises that “big” like tall is subjective).

Now let us move on to some basic definitions. In general, we will use lower case English letters to signify elements, upper case English letters to signify sets, and U to signify the special set, the universe.

Let U be a pre-specified well defined universe. If a is an element of the set S, then we shall write a Î A. The negation of this statement is, “a is not an element of the set S, and we shall write a Ï S.[2] So, the symbol “Δ is read as “belongs to,” “is in,” or “is a member of.” It is standard notation to use braces to enclose elements to signify a set. So, if one wishes to refer to the set consisting of the elements one, two, and three, then one would write, {1, 2, 3}. Also, it is standard notation to write a set with braces, use a variable to denote a generalised element of a set, and then describe the set thereafter based on axioms or previous definitions. We shall see an example of that later.

Example: Let U = {1, 2, 3, 4, 5}. Let A = {1, 3, 5}. Thus, 1 Î A, 2 Ï A, etc. Note every element in A must be in the universe; so, for the sake of this particular universe one could not discuss standard multiplication since , but 6 Ï U so it does not exist.

There are some special standard sets and symbols for them to denote sets that we use often. The sets under discussion are formulated from the real line (are either points of the line or are generalisations of the line. You knowledge of high school geometry will, no doubt, be of use in making concrete these abstract ideas that follow. As previously stated in the text, one of the most basic of sets is called the natural numbers. It has been with us since antiquity, and we will denote it as or IN = {1, 2, 3, 4, . . .} where the set never ends and includes all the whole or counting numbers that the student learnt in kindergarten or before. We shall denote the set of natural numbers along with zero as the set ^* = _À = {0, 1, 2, 3, 4, . . .}.[3] We shall denote the set {1} as ₁, the set {1, 2} as ₂, the set {1, 2, 3} as ₃ , and so forth so that

_k = {1, 2, 3, 4, . . ., (k - 1), k}. This definition is known as a recursive definition since we are inductively defining a myriad of sets at once; the three dots signify that the enumeration of the elements continues.[4] Likewise, we shall denote the set {0, 1} as , the set {0, 1, 2} as ,

the set {0, 1, 2, 3} as , and so forth so that = {0, 1, 2, 3, . . ., (p - 1), p}.

Another of the most basic of sets is called the integers. It has been with us quite a long time (the people of India invented the symbol of zero and were really the first to use it and negative numbers (in fact the number system that we use is of course the Hindu-Arabic number system since the Hindus created it, the Arabs adopted it and brought it west); it should also be noted that the Mayans also invented a zero independent of the Hindus). We will denote the set of integers as , ZZ , or Z such that ZZ = {0, 1, -1, 2, -2, 3, -3, 4, -4, . . .}.

Generalising from the integers, we have the rational numbers. The rationals are denoted as or Q such that = {x | x = where a Î ZZ , b Î ZZ , Ù b ¹ 0 }. This statement is read as the rationals are the set consisting of elements x such that x is equal to a divided by b where a is an integer, b is an integer, and b is not zero. Thus, the symbol “|” in this context means such that. Indeed, as I type this opus I am getting very tired of writing the words, “such that,” (I suppose it is an occupational hazard, I think we mathematicians are a lazy lot so we have invented many symbols to create a short hand to ease the amount of words necessary to communicate). A more general symbol for such that is, “',” and will be used liberally from this point onward. Typically it is not used in the notation inside the braces for a set only as a free-standing symbol. However, it is not incorrect to use it. Therefore, it is technically correct to write

= {x ' x = where a Î ZZ , b Î ZZ , Ù b ¹ 0 }.

We can simplify the definition of the rationals so that it does not have to depend solely on the integers. Note the definition = {x : x = where a Î ZZ , b Î } is logically equivalent to the previous definition of the rationals and in this case the colon means such that (which is the third of the standard notations for such that).

Generalising from the rational numbers, we have the real numbers. However, this is axiomatically executable, but practically most difficult to do in a basic introduction to sets. Therefore, we shall consider the set of reals from a geometric standpoint. The set of real numbers are denoted as , IR , or R such that = {x | x is a point on the line}. One could also define the reals from a sequential (or decimal) perspective by defining the reals to be

= {x | x is a number where x is an integer followed by a decimal and then a sequence of digits where each digit belongs to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}. Yes, this is a rather cumbersome definition, but one can prove that the sequential definition and the geometric definitions are equivalent. One other way to write is by writing (-¥ , ¥). The symbols ¥ and -¥ are not numbers, they simply represent that the line goes on ad infinitum to the left in the case of -¥ and to the right in the case of ¥.

Note, gentle reader that I skipped another standard set; that is because in a basic introduction to sets it is oft easier to ‘jump’ up to the reals, then return back to describe another set. That set is, of course, the irrationals. By the very nature of its name one can understand that it is composed of elements that are not rational. But, recall our discussion of the need for defining a domain of discourse or a universe. To say what something is not presupposes that everything has been specified! Putting it another way saying that the irrationals are numbers that are not rational is wrong since is a number that is not rational; but, it is also not irrational. Thus, the set of irrational numbers will be denoted as II , I , or I such that II = {x | x Î , x Ï Q}. So, the irrationals are the set of all real numbers that are not rational.

Note that this is a definition of something such that it is defined by what it is not. This definition by negation is oft quite useful; but one must understand what the first thing is (a real number) and the second thing is (a rational number) in order to understand what the third thing is (an irrational number) by way of what it isn’t.

Constructing sets from this perspective leaves us with the feeling that all is known and specified previously, but consider people before they thought of these sets. Consider the man or woman who first thought of these sets. Is it not rather astonishing to think that such was not known nor conceived, but someone thought of these ideas first? A facile way of considering the wonderful experience it must have been is by specify a set that is not contained within the set of reals. Laying aside the important principle of consideration of the specification of a universe for the moment, let us look at the idea of set from a construction standpoint. Note that we did this by specifying IN then ZZ then Q then IR . We deviated from it when we specified II .

Let us do it again. Define C or C to be the complex numbers such that

C = { x | x = a + bi where a Î R , b Î R , Ù i = }. Note that the complex numbers are really the plane (the horizontal axis consists of all points corresponding to the real part of the complex number and the vertical axis consists of all points corresponding to the i (‘imaginary’) part). So, why do so many people call complex number imaginary numbers when they correspond to things not so imagined but real? Indeed, if one argues that the reals correspond to real things and the complex numbers are ‘not real’ then why are both simply concepts corresponding to geometric forms (which recall are axiomatically given [point, line, and plane]). So, how real are the reals and imaginary are the complex numbers? But, I digress.[5]

Let U = N. Specify A = {x | 5 < x £ 11} in list form. Note the solution to this would be

A = {6, 7, 8, 9, 10, 11} . However, let U = R . Specify A = {x | 5 < x £ 11} in list form. Note one cannot produce a solution to this since there is no way to list out all the elements.

So, we can use the same symbol for two sets that are different (even though they seem quite the same, the difference in the universes made radical difference in the sets). Let us consider another example.

Let U = R . Consider B = {x | 5 < x £ 11, where x Î }. Consider C = {x | 5 < x £ 11, where x Î }. Note 6 Î B; 6 Î C; but, Î C; whereas, Ï B. So, what does it mean for two sets to be the same?

Let U be a well defined universe, A, B, C, and D be sets of elements from the well defined universe. The statement A = B means that every element of A is in B and every element of B is in A. There is a weaker notion that equality of sets. That notion is the idea of subset. A set C is a subset of D if for each element x in C, x is in D. When C is a subset of D we say D is a superset of C. C is a subset of D is denoted as C Í D or D Ê C (the same notation means D is a superset of C).[6] There is another notion between the idea of subset and equality, which is the notion of proper subsethood. A set B is said to be a proper subset of C if B Í C and B ¹ C. Denote B is a proper subset of C as B Ì C.[7] In rather a circular manner (granted) we can return to the definition of equality of two sets and state that two set A and B are equal, A = B, if and only if A Í B Ù