CHAPTER 1

Preliminaries

Math is a formal language useful in clarifying and exploring connections betweenconcepts. Like any language, it has a syntax that must be understood beforeits meaning can be parsed. We discuss the building blocks of this syntax in thischapter. The first is the variables that translate concepts into mathematics, andwe begin here. Next we cover groupings of these variables into sets, and thenoperators on both variables and sets. Most data in political science are ordered,and relations, the topic of our fourth section, provide this ordering. In the fifthsection we discuss the level of measurement of variables, which will aid us inconceptual precision. In the sixth section we offer an array of notation thatwill prove useful throughout the book; the reader may want to bookmark thissection for easy return. Finally, the seventh section discusses methods of proof,through which we learn new things about our language of mathematics. Thissection is the most difficult, is useful primarily to those doing formal theory ordevising new methods in statistics, and can be put aside for later reading orskipped entirely.

1.1 VARIABLES AND CONSTANTS

Political scientists are interested in concepts such as participation, voting, democracy,party discipline, alliance commitment, war, etc. If scholars are to communicatemeaningfully, they must be able to understand what one another isarguing. In other words, they must be specific about their theories and theirempirical evaluation of the hypotheses implied by their theories.

A theory is a set of statements that involve concepts. The statements compriseassumptions, propositions, corollaries, and hypotheses. Typically, assumptionsare asserted, propositions and corollaries are deduced from these assumptions,and hypotheses are derived from these deductions and then empiricallychallenged. Concepts are inventions that human beings create to help themunderstand the world. They can generally take different values: high or low,present or absent, none or few or many, etc.

Throughout the book we use the term "concept," not "variable," when discussingtheory. Theories (and the hypotheses they imply) concern relationshipsamong abstract concepts. Variables are the indicators we develop to measureour concepts. Current practice in political science does not always honor thisdistinction, but it can be helpful, particularly when first developing theory, tospeak of concepts when referring to theories and hypotheses, and reserve theterm variables for discussion of indicators or measures.

We assign variables and constants to concepts so that we may use them informal mathematical expressions. Both variables and constants are frequentlyrepresented by an upper- or lowercase letter. Y or y is often used to representa concept that one wishes to explain, and X or x is often used to represent aconcept that causes Y to take different values (i.e., vary). The letter one choosesto represent a concept is arbitrary—one could choose m or z or h, etc. There aresome conventions, such as the one about x and y, but there are no hard-and-fastrules here.

Variables and constants can be anything one believes to be important toone's theory. For example, y could represent voter turnout and x the level ofeducation. They differ only in the degree to which they vary across some setof cases. For example, students of electoral politics are interested in the gendergap in participation and/or party identification. Gender is a variable in the USelectorate because its value varies across individuals who are typically identifiedas male or female. In a study of voting patterns among US Supreme Courtjustices between 1850 and 1950, however, gender is a constant (all the justiceswere male).

More formally, a constant is a concept or a measure that has a singlevalue for a given set. We define sets shortly, but the sets that interest politicalscientists tend to be the characteristics of individuals (e.g., eligible voters), collectives(e.g., legislatures), and countries. So if the values for a given concept(or its measure) do not vary across the individuals, collectives, or countries, etc.,to which it applies, then the value is a constant. A variable is a concept or ameasure that takes different values in a given set. Coefficients on variables (i.e.,the parameters that multiply the variables) are usually constants.

1.1.1 Why Should I Care?

Concepts and their relationships are the stuff of science, and there is nothingmore fundamental for a political scientist than an ability to be precise in conceptformation and the statement of expected relationships. Thinking abstractly interms of constants and variables is a first step in developing clear theories andtestable hypotheses.

1.2 SETS

This leads us naturally into a discussion of sets. For our purposes, a set is justa collection of elements. One can think of them as groups whose members havesomething in common that is important to the person who has grouped themtogether. The most common sets we utilize are those that contain all possiblevalues of a variable. You undoubtedly have seen these types of sets before, asall numbers belong to them. For example, the counting numbers (0,1, 2,...,where ... signifies that this progression goes on indefinitely) belong to the setof natural numbers. The set of all natural numbers is denoted N, and anyvariable n that is a natural number must come from this set. If we add negativenumbers to the set of natural numbers, i.e., ..., -3, -2, -1, then we get the setof all integers, denoted Z. All numbers that can be expressed as a ratio of twointegers are called rational numbers, and the set of these is denoted Q. This setis larger than the set of integers (though both are infinite!) but is still missingsome important irrational numbers such as π and e. The set of all rational andirrational numbers together is known as the real numbers and is denoted R.

Political scientists are interested in general relationships among concepts.Sets prove fundamental to this in two ways. We have already discussed theassociation between concepts and variables. As the values of each variable, andso of each concept, are drawn from a set, each such set demarcates the rangeof possible values a variable can take. Some variables in political science haveranges of values equal to all possible numbers of a particular type, typicallyeither integers, for a variable such as net migration, or real numbers, for avariable such as GDP. More typically, variables draw their values from somesubset of possible numbers, and we say the variable x is an element of a subset ofR. For example, population is typically an element of Z₊, the set of all positiveintegers, which is a subset of all integers. (A + subscript typically signifiespositive numbers, and a — negative.) The size and qualities of the subset canbe informative. We saw this earlier for the gender variable: depending on theempirical setting, the sets of all possible values were either {Male, Female} or{Male}. The type of set from which a variable's values are drawn can alsoguide our theorizing. Researchers who develop a formal model, game theoreticor otherwise, must explicitly note the range of their variables, and we can useset notation to describe whether they are discrete or continuous variables, forexample. A variable is discrete if each one of its possible values can be associatedwith a single integer. We might assign a 1 for a female and 2 for male, forinstance. Continuous variables are those whose values cannot each be assigneda single integer. We typically assume that continuous variables are drawn froma subset of the real numbers, though this is not necessary.

A solution set is the set of all solutions to some equation, and may be discreteor continuous. For example, the set of solutions to the equation x² - 5x + 6 = 0is {2, 3}, a discrete set. We term a sample space a set that contains all of thevalues that a variable can take in the context of statistical inference. Whendiscussing individuals' actions in game theory, we instead use the term strategyspace for the same concept. For example, if a player in a one-shot game caneither (C)ooperate with a partner for some joint goal or (D)efect to achievepersonal goals, then the strategy space for that player is {C, D}. This will makesense in context, as you study game theory.

Note that each of these is termed a space rather than a set. This is not atypo; spaces are usually sets with some structure. For our purposes the mostcommon structure we will encounter is a metric—a measure of distance betweenthe elements of the set. Sets like Z and R have natural metrics. These examplesof sets form one-dimensional spaces: the elements in them differ along a singleaxis. Sets may also contain multidimensional elements. For example, a setmight contain a number of points in three-dimensional space. In this case, eachelement can be written (x, y, z), and the set from which these elements are drawnis written R³. More generally, the superscript indicates the dimensionality ofthe space. We will frequently use the d-dimensional space R^d in this book.When d = 3, this is called Euclidean space. Another common multidimensionalelement is an ordered pair, written (a, b). Unlike elements of R³, in which eachof x, y, and z is a real number, each member of an ordered pair may be quitedifferent. For example, an ordered pair might be (orange, lunch), indicating thatone often eats an orange at lunch. Ordered pairs, or more generally ordered n-tuples,which are ordered pairs with n elements, are often formed via Cartesianproducts. We describe these in the next section, but they function along thelines of "take one element from the set of all fruit and connect it to the set ofall meals."

Political scientists also think about sets informally (i.e., nonmathematically)on a regular basis. We may take as an example the article by Sniderman,Hagendoorn, and Prior (2004). The authors were interested in the source of themajority public's opposition to immigrant minorities and studied survey data toevaluate several hypotheses. The objects they studied were individual people,and each variable over which they collected data can be represented as a set. Forexample, they developed measures of people's perceptions of threat with respectto "individual safety," "individual economic well-being," "collective safety," and"collective economic well-being." They surveyed 2,007 people, and thus had foursets, each of which contained 2,007 elements: each individual's value for eachmeasure. In this formulation sets contain not the possible values a variablemight take, but rather the realized values that many variables do take, whereeach variable is one person's perception of one threat. Thus, sets here provideus with a formal way to think about membership in categories or groups.

Given the importance of both ways of thinking about sets, we will take sometime now to discuss their properties. A set can be finite or infinite, countableor uncountable, bounded or unbounded. All these terms mean what we wouldexpect them to mean. The number of elements in a finite set is finite; thatis, there are only so many elements in the set, and no more. In contrast, thereis no limit to the number of elements in an infinite set. For example, the setZ is infinite, but the subset containing all integers from one to ten is finite.A countable set is one whose elements can be counted, i.e., each one can beassociated with a natural number (or an integer). An uncountable set does nothave this property. Both Z and the set of numbers from one to ten are countable,whereas the set of all real numbers between zero and one is not. A boundedset has finite size (but may have infinite elements), while an unbounded setdoes not. Intuitively, a bounded set can be encased in some finite shape (usuallya ball), whereas an unbounded set cannot. We say a set has a lower bound ifthere is a number, u, such that every element in the set is no smaller than it,and an upper bound if there is a number, v, such that every element in theset is no bigger than it. These bounds need not be in the set themselves, andthere may be many of them. The greatest lower bound is the largest such lowerbound, and the least upper bound is the smallest such upper bound.

Sets contain elements, so we need some way to indicate that a given elementis a member of a particular set. A "funky E" serves this purpose: x [member of] A statesthat "x is an element of the set A" or "x is in A." You will find this symbol usedwhen the author restricts the values of a variable to a specific range: x [member of] {1, 2, 3}or x [member of] [0,1]. This means that x can take the value 1, 2, or 3 or x can be anyreal number from 0 to 1, inclusive. It is also convenient to use this notationto identify the range of, say, a dichotomous dependent variable in a statisticalanalysis: y [member of] {0,1}. This means that y either can take a value of 0 or a valueof 1. So the "funky E" is an important symbol with which to become familiar.Conversely, when something is not in a set, we use the symbol [??], as in x [??] A.This means that, for the examples in the previous paragraph, x does not takethe values 1, 2, or 3 or is not between 0 and 1. As you may have guessed fromour usage, curly brackets like {} are used to denote discrete sets, e.g., {A, B, C}.Continuous sets use square brackets or parentheses depending on whether theyare closed or open (terms we define in Chapter 4), e.g., [0,1] or (0,1), which arethe sets of all real numbers between 0 and 1, inclusive and exclusive, respectively.

Much as sets contain elements, they also can contain, and be contained by,other sets. The expression A [subset] B (read "A is a proper subset of B") impliesthat set B contains all the elements in A, plus at least one more. More formally,A [subset] B if all x that are elements in A are also elements in B (i.e., if x [member of] A,then x [member of] B). A [subset or equal to] B (read "A is a subset of B"), in contrast, allows A andB to be the same. We say that A is a proper subset of B in the first case butnot in the second. So the set of voters is a subset of the set of eligible voters,and is most likely a proper subset, since we rarely experience full turnout. Wealso occasionally say that a set that contains another set is a superset of thesmaller one, but this terminology is less common. The cardinality of a setis the number of elements in that set. Note that proper subsets have smallercardinalities than their supersets, finite sets have finite cardinalities, and infinitesets have infinite cardinalities.

A singleton is a set with only one element and so a cardinality of one. Thepower set of A is the set of all subsets of A, and has a cardinality of 2^|A|, where |A|is the cardinality of A. Power sets come up reasonably often in political scienceby virtue of our attention to bargaining and coalition formation. When oneconsiders all possible coalitions or alliances, one is really considering all possiblesubsets of the overall set of individuals or nations. Power sets of infinite sets arealways uncountable, but are not usually seen in political science applications.The empty set (or null set) is the set with nothing in it and is written 0.The universal set is the set that contains all elements. This latter concept isparticularly common in probability.

Finally, sets can be ordered or unordered. The ordered set {a, b, c} differsfrom {c, a, b}, but the unordered set {a, b, c} is the same as {c, a, b}. That is,when sets are ordered, the order of the elements is important. Political scientistsprimarily work with ordered sets. For example, all datasets are ordered sets.Consider again the study by Sniderman et al. (2004). We sketched four of thesets they used in their study; the order in which the elements of those sets ismaintained is critically important. That is, the first element in each set mustrefer to the first person who was surveyed, the second element must refer tothe second person, and the 1,232nd element must refer to the 1,232nd personsurveyed, etc. All data analyses use ordered sets. Similarly, all equilibriumstrategy sets in game theory are ordered according to player. However, thisdoes not mean all sets used in political science are ordered. For example, theset of all strategies one might play may or may not be ordered.

1.2.1 Why Should I Care?

Sets are useful to political scientists for two reasons: (1) one needs to understandsets before one can understand relations and functions (covered in thischapter and Chapter 3), and (2) sets are used widely in formal theory and in thepresentation of some areas of statistics (e.g., probability theory is often developedusing set theory). They provide us with a more specific method for doingthe type of categorization that political scientists are always doing. They alsoprovide us with a conceptual tool that is useful for developing other importantideas. So a basic familiarity with sets is important for further study.