CHAP. 2] RELATIONS 25
EXAMPLE 2.3
(a) A=( 1 , 2 , 3 )andB={x, y, z}, and letR={( 1 , y), ( 1 , z), ( 3 ,y)}. ThenRis a relation fromAtoBsinceR
is a subset ofA×B. With respect to this relation,1 Ry, 1 Rz, 3 Ry, but 1Rx, (^2) Rx, (^2) Ry, (^2) Rz, (^3) Rx, (^3) Rz
The domain ofRis{ 1 , 3 }and the range is{y, z}.
(b) Set inclusion⊆is a relation on any collection of sets. For, given any pair of setAandB, eitherA⊆B
orA⊆B.
(c) A familiar relation on the setZof integers is “mdividesn.” A common notation for this relation is to write
m|nwhenmdividesn.Thus6 | 30 but 7|25.
(d) Consider the setLof lines in the plane. Perpendicularity, written “⊥,” is a relation onL. That is, given any
pair of linesaandb, eithera⊥bora⊥b. Similarly, “is parallel to,” written “||,” is a relation onLsince
eithera‖bora‖b.
(e) LetAbe any set. An important relation onAis that ofequality,
{(a, a)|a∈A}
which is usually denoted by “=.” This relation is also called theidentityordiagonalrelation onAand it will
also be denoted byAor simply.
(f) LetAbe any set. ThenA×Aand∅are subsets ofA×Aand hence are relations onAcalled theuniversal
relationandempty relation, respectively.
Inverse Relation
LetRbe any relation from a setAto a setB. TheinverseofR, denoted byR−^1 , is the relation fromBtoA
which consists of those ordered pairs which, when reversed, belong toR; that is,
R−^1 ={(b, a)|(a, b)∈R}
For example, letA={ 1 , 2 , 3 }andB={x, y, z}. Then the inverse of
R={( 1 , y), ( 1 , z), ( 3 ,y)} is R−^1 ={(y, 1 ), (z, 1 ), (y, 3 )}
Clearly, ifRis any relation, then(R−^1 )−^1 =R. Also, the domain and range ofR−^1 are equal, respectively, to
the range and domain ofR. Moreover, ifRis a relation onA, thenR−^1 is also a relation onA.
2.4Pictorial Representatives of Relations
There are various ways of picturing relations.Relations on R
LetSbe a relation on the setRof real numbers; that is,Sis a subset ofR^2 =R×R. Frequently,Sconsists
of all ordered pairs of real numbers which satisfy some given equationE(x, y)=0 (such asx^2 +y^2 =25).
SinceR^2 can be represented by the set of points in the plane, we can pictureSby emphasizing those points
in the plane which belong toS. The pictorial representation of the relation is sometimes called thegraphof the
relation. For example, the graph of the relationx^2 +y^2 =25 is a circle having its center at the origin and radius 5.
See Fig. 2-2(a).