Chapter 5 Relations

Relations are the essence of knowledge What is important in science is not knowledge of particulars but knowledge of the relations among phenomena. Educational scientists can “know” about achievement only as they study achievement in relation to nonachievement and in relation to other variables.

Relations as Sets of Ordered Pairs Relations in science are always between classes or sets of objects Consider the two sets as one set of pairs. Then this set is a relation. Figure 5.1 A relation is a set of ordered pairs. Any relation is a set, a certain kind of set: a set of ordered pairs.

Relations as Sets of Ordered Pairs When discussing relations, there are two special types of sets that play an important role. One set is called the domain and the other is called the range. Defining the domain and range in a relation is important because they play a key role in defining a function. A function can be thought of as a special kind of relation. A relation is a function when each element of the domain is paired with one and only one member of the range.

Determining Relations in Research There is another way to define a relation that may help us. Let A and B be sets. If we pair each individual member of A with every member of B, we obtain all the possible pairs between the two sets. This is called the Cartesian product of the two sets and is labeled A × B. A relation is then defined as a subset of A × B; that is, any subsets of ordered pairs drawn from A × B is a relation. Figure 5.3

Determining Relations in Research There are many subsets of pairs of A × B, most of which do not “make sense” or do not interest us. The relation of “marriage” is a method or procedure for distinguishing married couples from all possible pairings of men and women. No matter what sets of ordered pairs we choose. It is a relation. It is up to us to decide whether or not the sets we pick make scientific sense according to the dictates of the problems to which we are seeking answers. Almost all science pursues and studies relations. A “relationship” is not a “relation.”

Rules of Correspondence and Mapping Any objects—people, numbers, gambling outcomes, points in space, symbols, and so on—can be members of sets and can be related in the ordered-pair sense. A rule of correspondence is a prescription or a formula that tells us how to map the objects of one set onto the objects of another. Figure 5.4. In a relation the two sets whose “objects” are being related are called the domain and the range, or D and R. The rule of correspondence says: If the object of D is female assign a “0,” if male assign a “1.”

Some ways to Study Relations Graphs A graph is a drawing in which the two members of each ordered pair of a relation are plotted on two axes, X and Y. Tables Table 5.1, 5.2, 5.3 Graphs and Correlation Research Examples

Multivariate Relations and Regression Some Logic of Multivariate Inquiry In logic, “If p, then q” is called a conditional statement, and it is possible to conceptualize most research problems and study the structure of scientific arguments using conditional and related statements. Contemporary researches are more likely to say “If p, then q, under conditions r and t.” This conditional statement can be written: p→q ∣ r, t (“ ∣ ” means “under conditions,” or “given”) Or, somewhat simpler, we can write: (p1,p2,p3) →q which means “If p1 and p2 and p3, then q.” The simplest way to show the relations graphically is with so-called path diagrams.

Multiple Relations and Regression Figure 5.9, 5.10 We say, If x, then y, or: If x1, x2, x3, then y. Many authors talk about “causal analysis,” especially when talking about problems such as those given in Figure 5.9 and Figure We prefer to avoid the words cause and causal because they are exceedingly sticky ideas—for instance, what is a cause? —and because their use is not necessary.