RELATIONS

Everyone is familiar with relations as a connection between two or more things; relations permeates everyday life and conversation and are not peculiar to mathematics. It is common to hear reference to such relations as "is the father of," "is the brother of," "is married to," "is to the left of," "is above of," "is smaller than," "is taller than," "is heavier than," "is between," and so on. Elementary Geometry is concerned with such relations as "is similar to," "coincides with," "is congruent to," "is perpendicular to," "is parallel to," and "is collinear with." In the arithmetic of numbers the interest is in such relations as "is equal to," "is greater than," "is less than," "is the sum of," and "is the product of."

Relations may be classified according to number of terms that are related and is called the degree of the relation. It is customary in mathematics to use a terminology derived from Greek rather than Latin to classify relations. A relation connecting two terms is called a dyadic relation and one connecting three terms a triadic relation. Similarly, there are tetradic (four terms), pentadic (five terms), and polyadic (many terms) relations. The most common relations, in general and in mathematics, are the dyadic relations, as may be seen in the list above. The relation "is between" is an example of a triadic relation in the form "a is between b and c." Other triadic relations in the above list are "is collinear with," "is the sum of," and "is the product of," when used in the form "a is collinear with b and c," "a is the sum of b and c," and "a is the product of b and c." Higher degree relations are less common. Degree is the most fundamental characteristic of relations. All other properties of relations presuppose that of degree.

Because of the wide occurrence of dyadic relations, the general properties of these relations have been given distinctive names. There are three classes of general properties of dyadic relations, namely, reflexivenss, symmetry, and transitivity. In the following discussion a dyadic relation will be symbolized by R and the terms that it connects by a, b, c, and so on. If the terms a and b are related by the relation R, it shall be written aRb; if a is not related to b by the relation R, it will be written ~(aRb).

1. A dyadic relation R is said to be reflexive, if it is true that aRa; that is, if the relation R holds with respective to itself. For example, the relation "is equal to" is reflexive when applied to numbers, since for any number a = a: 2 = 2, 40 = 40. But the relation "is the father of" is not reflexive, since no person is a father of himself. Such dyadic relations are said to be irreflexive as are the relations "is greater than" and "is less than" when applied to numbers. When a dyadic relation is sometimes reflexive and sometimes irreflexive, they are said to be nonreflexive. That is, a relation is nonreflexive relation if the relation possibly but not necessarily combines a term with itself. Such relations as "admires," "likes," and "hurts," when applied to persons, is nonreflexive, since a person may or may not admire, like, or hurt himself.

2. A dyadic relation R is said to be symmetric, if it is true that if aRb, then bRa. For example, the relation "is equal to," when applied to numbers, and the relation "is married to," when applied to people, are symmetric relations. For if a and b are two numbers and if a = b, then b = a, then the relation "is equal to" is symmetric. Similarly, if a and b are two people, a male and female, and if a is married to b, then b is married to a. A dyadic relation R is said to be asymmetric if aRb, then ~(bRa); that is, if the relation between a and b does not hold between b and a. For example, the relation "is greater than," when applied to numbers, is asymmetric, because if a and b are two numbers and a > b, it will not be true that b > a. Similarly, the relation "is father of," when applied to people, is asymmetric, because if a and b are two people and if a is the father of b, it is not true that b is the father of a. But if the dyadic relation R is sometimes symmetric and sometimes not symmetric, then the relation R is said to be nonsymmetric. That is, the dyadic relation R is nonsymmetric, if aRb may, but need not imply that bRa. For example, the relations "is brother of" and "admires," when applied to people, are nonsymetric, because if a and b are two people, and if a is the brother of b, then b may or may not be the brother of a; b may be the sister of a. Similarly, if a and b are two persons, then if a admires b, b may or may not admire a.

3. A dyadic relation R is said to be transitive, if aRb and bRc, then aRc. An example of a transitive relation is the relation "is equal to," when applied to numbers. So is the relation "is greater than," when applied to number, and the relation "is included in," when applied to sets, and the relation "implies," when applied to propositions. A dyadic relation R is said to be intransitive, if aRb and bRc, then ~(aRc). For example, if a, b, and c are people, and the relation R is the relation "is the father of," a is the father of b and b is the father c, it is not true that a is the father of c; a is the grandfather of c. A dyadic relation R is said to be nontransitive, if the relation is sometimes transitive and sometimes not transitive. That is the relation of a to c is possible, but not implied. An example is the dyadic relation "admires," when applied to people. If a admires b, and b admires c, then a may or may not admire c.

These three properties are totally independent of each other. That is to say, any one property of a dyadic relation does not imply any the other property. For example, the relation "is contemporaneous to" is reflexive, symmetric, and transitive, when applied to people; the relation "is ancestor of," when applied to people, is irreflexive, asymmetric, and transitive; the relation "is married to," when applied to people, is irreflexive, symmetric, and intransitive; the relation "admires," when applied to people, is nonreflexive, nonsymmetric, and nontransitive; the relation "is the father of," when applied to people, is irreflexive, asymmetric, and intransitive.

There is one dyadic relation that is of paramount importance in mathematics. This is the so-called equivalence relation, which is reflexive, symmetric, and transitive. That is to say, a dyadic relation is an equivalence relation, ≡, if and only if the following three axioms or postulates hold for the relation:

E-1.	(reflexive):	a ≡ a.
E-2.	(symmetric):	if a ≡ b, then b ≡ a.
E-3.	(transitive):	if a ≡ b and b ≡ c, then a ≡ c.

Examples of equivalence relation are abundant. Some of these are "is contemporaneous with" and "is the same age," when applied to people; "is equal to," when applied to number; "is similar to," and "is congruent to," when applied to triangles in geometry; and so on. These dyadic relations have a common algebraic structure defined by the above axiom set. Incidentally these examples, since they are interpretations of this set of axioms, establish the consistency of the postulate set. It is evident that the concept of equivalence relation is a generalization of the equals relation, in the sense of identity.

In addition to the equivalence relation there is another dyadic relation that is of fundamental importance in mathematics. This is the order or series relation which is irreflexive, asymmetric, and transitive. That is to say, a dyadic relation R is an order relation if and only if the following axioms or postulates hold for the relation:

O-1.	(irreflexive):	~(a R a).
O-2.	(asymmetric):	if (a R b), then ~(b R a).
O-3.	(transitive):	if (a R b) and (b R c), then (a R c).
O-4.	(determinate):	if (a ≠ b), then (a R b) or (b R a), but not both.

Examples of order relation are abundant. Some of these are "is to left of," "is to the right of," "is above," "is below," "is after," "is before," when applied to special elements, and "is greater than" (>), "is less than" (<), when applied to numbers. If a < b, then b > a. That is, b > a is just another way of saying that a < b.

The triadic relations "is the sum of" and "is the product of," when used in the form "c is the sum of a and b," and "c is the product of a and b," have a different set of properties from the properties of the dyadic relations. The binary operator, symbolized by ⊕, which relates two elements a and b of the set G to a third element c in G, forming a triadic relation between the elements, has the following five properties. (The symbol = is taken as denoting an equivalence relation.)

G-1.	(closure):	a ⊕ b = c is a unique member in G.
G-2.	(commutative):	a ⊕ b = b ⊕ a.
G-3.	(associative):	(a ⊕ b) ⊕ c = a ⊕ (b ⊕ c).
G-4.	(identity):	There exists a unique identity element e in G such that a ⊕ e = a.
G-5.	(inverse):	If a is any element of G, there exists a unique inverse element a´ in G such that a ⊕ a´ = e.

When these properties are taken as the axioms or postulates for some set G they define a mathematical system called a group and the symbol ⊕ is taken as an operator having the properties specified by the axioms or postulates and the symbol = as an equivalence relation. Some groups who do not have a commutative postulate or axiom are called non-commutative groups or non-Abelian groups; those groups having the commutative axiom are called commutative groups or Abelian groups, named for the Norwegian mathematician Niels Henrik Abel (1802-1829). Groups are assumed not to be commutative unless otherwise specified. The commutative property of the triadic relation is similar to the symmetric property of dyadic relation; it is really the symmetric property of the operator that relates two elements to a third element to form the triadic relation between them. The identity property says that there is only one element in the set G, called the identity element, such that the operator which relates any element in G with the identity element leaves that element identically the same and unchanged; that is, this triadic relation connects any element in the set G to itself by the identity element. This identity property is similar to the reflexive property of dyadic relation. The inverse property says that for every element in the set G there is another element in G, its inverse, by which the operator "reduces" that element to the identity element; that is, the triadic relation, which relates any element to its inverse, relates them to the identity element. A group is the simplest mathematical system because it contains only one operator. The following theorems hold for groups with respect to the operator ⊕:

T-1.	The identity element e of a group is unique.
T-2.	The inverse element a´ of an element a is unique.
T-3.	If a ⊕ b = a ⊕ c, then b = c (the cancellation law).
T-4.	If a and b are any elements of G, then (a ⊕ b)´ = a´ ⊕ b´.

Examples of groups: The most important examples of groups are groups of numbers.

1. The integers form an infinite Abelian group with respect to the operation of addition.

   G-1.	The sum of two integers is an integer.
   G-2.	For any integers a and b, a + b = b + a.
   G-3.	For any integers a, b, and c, (a + b) + c = a + (b + c).
   G-4.	The integer 0 is the identity element of the group, since a + 0 = 0 + a = a for any integer a.
   G-5.	The integer -a is the inverse of any integer a, since a + (-a) = (-a) + a = 0.

   Note that the integers under addition form an Abelian Group, since it has the commutative property G-2. Also note that the natural numbers are not a group with respect to addition (no identity and no inverse) or with respect to multiplication (no inverse except for the number 1). The whole numbers are not a group with respect to addition (no inverse) or with respect to multiplication (no inverse except for the number 1).

2. The rational numbers under addition form an infinite Abelian group.

3. The non-zero rational numbers under multiplication form an infinite Abelian group with identity element 1 and the inverse of each number is its reciprocal.

4. The real numbers form groups with respect to addition and multiplication, but not with respect to subtraction or division.

5. The smallest group consists of just one element, number 0 in the set {0}, with addition operator:

   G-1.	0 + 0 = 0 is a unique element in {0}.
   G-2.	0 + 0 = 0 + 0.
   G-3.	(0 + 0) + 0 = 0 + ( 0 + 0).
   G-3.	Since 0 + 0 = 0, the number zero is the identity element.
   G-4.	Since 0 + 0 = 0, the number zero is its own inverse element (0 = -0).