Relations and Cartesian Products

We have seen that any property of objects (belonging to some given domain) determines a set: the set of all objects (in the given domain) that have the property. We can therefore use sets as substitutes for properties. (By doing so we disregard the difference between any two properties that determine the same set.)

There are creatures that, like properties, are true of objects, but which involve more than one object: they relate objects to each other. For example, the parent-child relation holds for any pair of objects, x and y, such that x is a parent of y. Set theory provides a very simple and elegant way of representing these creatures:

Regard the relation as a property of ordered pairs and represent it, accordingly, as a set of ordered pairs.

Thus, the parent-child relation is the set of all ordered pairs (x, y), such that x is a parent of y. If, for the sake of illustration, we restrict our universe to a domain consisting of:

Olga, Mary, Ruth, Jack, John, Abe, Bert, Nancy, Frieda,

and if the parent-child relation among these people is given by:

Abe is the father of Ruth and Jack,

Olga is the mother of Mary, Abe and Nancy, Jack is the father of Bert,

John is the father of Nancy,

and there are no other parent-child relationships, then–over this domain–the parent-child relation is simply the set:

{(Abe, Ruth), (Abe, Jack) , (Olga, Mary), (Olga, Abe), (Olga, Nancy), (Jack, Bert), (John, Nancy)}

Note that the child-parent relation is obtained by switching the two coordinates. It contains as members: (Ruth, Abe), (Jack, Abe), (Mary, Olga), etc.

Relations that involve three members are construed, accordingly, as sets of 3-tuples. For example, the betweenness-relation–which holds between any three points x, y, z on a line such that y is between x and z–is the set of all triples (x, y, z) such that y is between x and z. Here, to sum up, are some basic notions and terms:

A binary relation is a set of ordered pairs.

An n-ary relation (also called an n-place relation) is a set of n-tuples.

Unqualified ‘relation’ means often a binary relation.

If R is an n-ary relation, then n is referred to as the arity of R, or the number of places of R.

{(x¹, x2, . . . , xn) : . . . x1. . . x2. . . xn. . .} is the set of all tuples (x¹, x2, . . . , xn) satisfying the condition stated by ‘. . . x1, . . . x2, . . . xn. . ..

The betweenness relation above can be written as:

{(x, y, z) : y is between x and z}

where ‘x’ ‘y’ and ‘z’ range over geometrical points. Here are some other examples:

{(x, y) : x is a parent of y}, {(x, y) : y is a parent of x}, {(x, y, z) : x introduced y to z}

{(x, y) : x and y are real numbers and y = 2x + 1}

{(x, y) : x and y are natural numbers and x ≥ y}

Note: The variables in relational notation are used as place holders, that is, to correlate coordinates with places in the defining expression. Different variables, or the same variables in different roles, can achieve the same effect:

{(x, y) : x is a parent of y} = {(y, x) : y is a parent of x} = {(u, x) : u is a parent of x}

But

{(x, y) : x is a parent of y} 6= {(x, y) : y is a parent of x}

The first relation consists of pairs in which the parent occupies the first coordinate, the child–

the second; in the other relation the child is in the first place, the parent–in the second.

Self-explanatory variants of our notation involve repetitions of variables. e.g., {(x, x) : x ∈ D}

is the set of all pairs (x, x), where x ranges over D. It is equal to{(x, y) : x, y ∈ D and x = y}.

Note: The arity of the relation is the length of the tuple; it may be greater than the number of different variables that appear in the definition, because, as we have just seen, the same variable can occupy different places in the tuple.

Relations Over a Given Domain: Often, we consider relations that relate objects of particular kinds: numbers, people, animals, words, etc. We say that a relation is over D if it consists of tuples whose members belong to D.

Usually, the variables range over well defined domains. In ‘x is an uncle of y’, ‘x’ and ‘y’

range, obviously, over people. Relations can, however, relate objects of different kinds; e.g., the ownership relation that holds between x and y, just when x is a person, and y is an object owned by x.

Homework

5.4 Consider binary relations consisting of the pairs (x, y), determined respectively by fol-lowing conditions. (When the signs≥, <, =, 6= are used, the variables range over the natural numbers.)

(1) x is a brother of y. (2) y is a sibling of x. (3) x ≥ y, (4) y < x, (5) x⊆ y, where x and y are sets of natural numbers. (6) x6= y (7) y = x (8) x is an

ancestor of y. (9) y is a child of x. (10) x = y = 3 (11) x and y are natural numbers.

Find out which of the following inclusions is true (the numbers refer to the corresponding relations). Justify your answers.

(1)⊆ (2) (2)⊆ (1) (3)⊆ (4) (4)⊆ (3) (7)⊆ (3) (7)⊆ (4) (7)⊆ (10) (10)⊆ (7) (8)⊆ (9) (9)⊆ (8)

5.5 A (binary) relation, R, is:

symmetric, if whenever (x, y)∈ R, also (y, x) ∈ R, transitive, if whenever (x, y), (y, z)∈ R, also (x, z) ∈ R, reflexive over the domain D, if whenever x∈ D, (x, x) ∈ R.

Note that reflexivity depends also on the domain. A relation which is reflexive over a domain can cease to be so with respect to a larger domain. Usually, when we speak of a reflexive relation, the domain is presupposed.

Find out which of the relations of 5.4 is symmetric, which is transitive and which is reflexive (over the naturally associated domain). Justify your conclusions. If the property in question does not hold show this by a counterexample.

Cartesian Products

Let X1, X2, . . . , Xn be sets. The Cartesian product of X1, X2, . . . , Xn, denoted as:

X1× X²× . . . × Xⁿ,

is the set of all n-tuples in which the first coordinate is a member of X1, the second coordinate is a member of X2, and so on ..., the n^th coordinate is a member of Xn. Formally, for all x:

x∈ X¹× . . . × Xⁿ iff there are x1, x2, . . . , xn such that: x = (x1, x2, . . . , xn) and xi ∈ Xⁱ, for i = 1, 2, . . . , n.

We can also express this using the notation for sets of tuples:

X1× X²× . . . × Xⁿ = {(x¹, x2, . . . , xn) : xi ∈ Xⁱ, for i = 1, 2, . . . , n}

Note: In ‘(x1, x2, . . . , xn)’, the index is the place-number in the sequence. But there is no general rule that ties indices to place-numbers. The first member of (x3, x1, x1) is x3, the second is x1 and the third is x1.

Examples:

{1, 2} × {0, 1, 2} = {(1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)}

{1, 2}×{1, 2}×{2, 3} = {(1, 1, 2), (1, 1, 3), (1, 2, 2), (1, 2, 3), (2, 1, 2), (2, 1, 3), (2, 2, 2), (2, 2, 3)}

Cartesian Powers: If Xi = X, for i = 1, . . . n, then X1× . . . × Xⁿ is said to be the n^th Cartesian power of X and is denoted as:

Xⁿ

Obviously, Xⁿ is the set consisting of all n-tuples of members of X. If n = 2, it is the set of all ordered pairs of members of X.

Historically, the concept of Cartesian product was derived from geometry. A coordinate system for the plane consists of two perpendicular directed lines, which are referred to as axes. Each point in the plane can be projected on the two axes, determining thereby an ordered pair of real numbers (x, y), where x represents the projection on the first axis, y–the projection on the second. Vice versa, every pair of numbers determines a unique corresponding point. In this way it is possible to identify the plane with the Cartesian product R×R, or R², where R is the set of all real numbers. Similarly, the three dimensional space can be identified with R³. This representation, by now a commonplace, amounts to a major breakthrough in the history of science. It was discovered around 1637 by Descartes. (‘Cartesian’ derives from his Latin name ‘Cartesius’.)

All of geometry is reducible, in principle, to a system that deals with pairs, or triples of numbers. Moreover, the concept of a Cartesian product makes possible the definition and the study of higher dimensional geometrical spaces, structures that resists visualization. A geo-metrical space of 4 dimensions is simply, R⁴, that of 5-dimensions is R⁵, and an n-dimensional space is Rⁿ.

Homework

5.6 Let X1 = {0, 2, 4}, X² = {0, 5}. Write down the following sets in the curly bracket notation.

X1× X², X2× X¹, X₁², X₂³ X1× {2} × X¹ X1× {∅} × X¹, X1× ∅ × X¹ . 5.7 Prove the following:

(i) X× (Y ∪ Z) = (X × Y ) ∪ (X × Z) (ii) (X∪ Y ) × Z = (X × Z) ∪ (Y × Z)

(iii) X1× . . . × Xⁿ =∅ iff one of X¹, X2, . . . , Xn is empty.

5.8 Prove that, if no Xiis empty, then X1× X²× . . . × Xⁿ= Y1× Y²× . . . × Yⁿ iff Xi = Yi, for i = 1, 2, . . . , n.

What can you deduce from this, concerning the equality X × Y = Y × X ?

Functions

Historically, functions have been conceived as laws by which a magnitude is determined by another; for example, the distance traveled by a falling body is said to be a function of the time of fall. Functions have been also considered as rules that correlate objects with objects.

Thus, there is a function that correlates with every number, x, the number x²; and one that correlates with x the number 2x−1. Function’s can be defined for any kind of objects; e.g., there is a function that correlates the person’s mother with each person, and there is one that correlates, with each star, the galaxy it belongs to.

Commonly ‘f (x)’ denotes the object that the function f correlates with x (assuming, of course, that f is defined for x). We say that f (x) is the value of f for the argument x, and also that it is the value of x under f .

The intuitive concept of “rule”, or “law”, by which the correlation is determined, is too vague for mathematical purposes. Historically, the list of entities admitted as functions kept growing, until mathematicians came to realize that they need an abstract concept of function, which does not rely on the notion of a defining rule. Set theory provides a perfect definition of such a concept. Consider the set of all pairs (x, y) such that y is the value correlated with x. One can define the function as being simply this set. Given such a set, the function assigns a value to each x for which there is a y such that (x, y) is in the set; that y is the value of x under the function.

Not every set of ordered pairs will do as a function. It should satisfy the condition that, for every x, there is at most one y such that (x, y) is in the set (the function should assign to any object no more than one value). Any set satisfying this condition is a function. Stated in full, the definition is this:

A function, f , is a relation (set of ordered pairs) such that: for all x, y, y⁰, if (x, y)∈ f and (x, y⁰)∈ f, then y = y⁰.

If there is a y such that (x, y) ∈ f, then we say that f is defined for x. The domain of the function, which we denote as dom(f ), is the set of all objects for which the function is defined.

If x ∈ dom(f), then the value of f for x is the unique y such that (x, y) ∈ f. The value is denoted by ‘f (x)’.

Example If f ={(x, 3x² + 1) : x∈ N}, where N is the set of natural numbers, then f is a function, dom(f ) = N , and f (x) = 3x²+ 1 for all x∈ N

Functions differ just when they differ as sets of ordered pairs. It is easy to see that two functions, f and g, are equal just when they are defined for the same objects and assign to

every object the same value. Formally:

f = g iff dom(f ) = dom(g) and f (x) = g(x), for all x∈ dom(f) .

Functions come under many names, which suggest diverse aspects and uses of the concept.

We have correlation, assignment, and correspondence, which suggest a pairing of objects with objects. And we have also operator and operation, which suggest the transforming objects into objects. We might say that squaring is an operation by which any number x is transformed into x². Transformation is itself a term adopted in mathematics for functions of a certain type. We have also the term mapping, which suggests both a matching of items and a copying of one thing to another.

One-to-One Functions and Equinumerous Sets: A function f is said to be one-to-one if it correlates with different objects in its domain different values; that is, for all x, y∈ dom(f):

x6= y =⇒ f(x) 6= f(y) Or, equivalently expressed:

f (x) = f (y) =⇒ x = y

The concept of a one-to-one function is extremely important in mathematics. It serves, among other things, to define the concept of equinumerous sets; these are sets that have the same number of members:

The set X is equinumerous to the set Y if there exists a one-to-one function f , such that X = dom(f ) and Y = {f(x) : x ∈ X}. (In words: Y is the set of all objects correlated via f with members of X.)

Little reflection will show that this definition captures all there is to “having the same num-ber”. There are exactly as many forks in the drawer as there are spoons, just when one can pair with every fork a spoon, so that different forks are paired with different spoons and every spoon is paired with some fork. The function that correlates with each fork its paired spoon is the function that satisfies the conditions of the last definition. Vice versa, any such function determines a pairing of spoons with forks.

A crucial feature of the definition is that it applies to all sets, finite as well as infinite. We can therefore define when two infinite sets have the same number of elements. The definition was introduced by Cantor who derived from it the general concept of a cardinal number; that is, a (possibly infinite) number, which can serve as an answer to the question: How many elements are there in a set?

Functions of Several Arguments: So far we have discussed functions of one argument, i.e., functions that correlate objects with objects Often, however, we correlate objects with more than one object. We may correlates with every two numbers, x and y, their sum: x + y;

and we may correlates with x, y and z, the number x^y + z. Such cases are construed as functions of many arguments.

It is possible to subsume functions of n arguments under functions of one argument, by regarding them as one-argument functions defined for n-tuples. Under this construal, the function that assigns to x and y the difference x − y is a one-argument function, whose domain is R× R; it assigns to any ordered pair the number obtained by subtracting the second coordinate from the first.

Alternatively–and this is sometimes more convenient–we can define a function of n argu-ments as an (n+1)− ary relation, say f, which satisfies the condition:

If (x1, . . . , xn, y)∈ f and (x¹, . . . , xn, y⁰)∈ f, then y = y⁰

An n-place function, f , is defined for x1, . . . , xn, if there exists y, such that (x1, . . . , xn, y)∈ f.

Such a y is, of course, unique and we denote it by:

f (x1, . . . , xn)