i
D e f i n i t i o n 3 .2 .1 ( D i s t a n c e F u n c t i o n ) .4 d is ta n c e fu n c tio n (o r d istan ce ,/ d on a n o n
em p ty set X is a fu n c tio n su c h that d : X x X — > R-*" U {0}. T h e n o ta tio n d ( p . q ) is read
i
5 as th e d ista n c e from p to q.
\
G iven a n o n e m p ty se t X . a n y function d : X x A' —>■ [0 . oc) im p o ses a n o tio n of .
" a b s tra c t" d ista n c e on th e p o in ts o f X . We use th e w ord " a b s tra c t" b ec a u se it m ay n o t be «
im m e d ia tely clear th a t a n a r b it r a r y n o n n eg ativ e real-v alu ed fu n c tio n on X x A' satisfy in g no p a rtic u la r axiom s n ec essarily d e sc rib e s w h a t can b e re g a rd e d o n a n in tu itiv e level as a "d istan ce." For ex am p le, it m ay n o t a p p e a r re aso n ab le to allow d ( p . p ) > 0 for so m e point
w ould e x p e c t a fu n c tio n d : X x A' —> [0 . 3c ) to sa tis fy if d is to d esc rib e a “re aso n ab le" n o tio n o f d is ta n c e a m o n g p o in ts of X . In T a b le 3.1. th e fo u r m o st co m m on m e tric ;ixiom s a re liste d . L et d b e a d is ta n c e fu n c tio n on X a n d Vp. q. r 6 X : ! M l d ( p . p ) = 0 1 1 M 2 d ( p . q ) < d ( p . r ) + d ( r . q ) M 27. c/(/>. r/) < m a x {d (p . r ) . d ( r . q ) } M 3 d( p . q ) = d( q. p) M 4 d( p . q ) = d( q . p ) = 0 = > p = q M 47. d ( p . q ) =1) = > p = q
T able 3.1: B asic M e tric A x io m s
A xiom s M 2 7 a n d M 47 are th e "stro n g '' versions o f a x io m s M 2 an d M 4 respectively. N o te t h a t a x io m M l re q u ire s th a t th e d is ta n c e from a p o in t to its e lf be 0. a q u a lity know n
! as reflexivity. A x io m M 2 effectively tells us t h a t w h en we w ish to "m ove” from o n e point
p to a n o th e r p o in t q. th e re is no a d v a n ta g e (fro m th e p o in t o f view o f m inim izing d ista n c e )
in "v isitin g " so m e o t h e r p o in t o f X along th e way. T h is is c o m m o n ly know n as th e tria n
gle inequality. F ro m a d iffe ren t p ersp ectiv e, it also g u a r a n te e s t h a t d( p. q) re p re se n ts th e
m in im u m effort to "m ove" from p to q. A xiom M 3 is k n o w n as sy m m e try . If th e re ex ists a s u b se t o f X in w h ic h th e d ista n c e b etw een each p a ir o f p o in ts is 0. axiom M 4 identifies th e s u b s e t w ith a sin g le re p re se n ta tiv e p o in t, a q u a lity t h a t is so m etim e s called id e n tity o f
in d isc e m ib le s [58].
. D e f i n i t i o n 3 .2 .2 ( P s e u d o - Q u a s i m e t r i c ) L et d be a d ista n c e fu n c tio n on X . d is a pseu-
d o -q u a sim e tric a n d ( X . d ) is a p se u d o -q u a sim c tric sp ac e i f a x io m s M l and M 2 are satisfied.
i D e f i n i t i o n 3 .2 .3 ( P s e u d o - M e t r i c ) Let d be a d ista n ce fu n c tio n on X . d is a pseudo-
>
t. m etric a n d ( X . d ) is a p se u d o -m e tric sp ac e i f a xio m s M l . M 2 , a n d M 3 are satisfied.
D e f i n i t i o n 3 .2 .4 ( M e t r i c ) Let d be a d istance fu n c tio n on X . d is a m e tric a n d ( X . d )
is a m e tric sp a c e i f a x io m s M l . M 2. M 3, an d M 4 are sa tisfied .
E x a m p l e 3 .2 .5 L et
E
d e n o te the. set o f real n u m b ers a n d the set. { 1 . 2 . 3 . . . . } o f positivei L.
10
integers. For each n € Z ~ . d efin e d rl : R" x R" —r [0. o c ) by
d n ( ( x i x „ ) . (i j i y n )) =
\
y > , - in )1 1=1
.Vote that d\ describes th e usual d ista n ce between real n u m b e rs on the n u m b er line. <l> the. usual d ista n ce betw een p o in ts in the C a rte s ia n p la n e , a n d d \ th e u su a l d istance between points in th re e -d im e n sio n a l E u c lid e a n sp a c e . The distance fu n c tio n d n is a m e tric , generally know n as the E u c lid e a n d is ta n c e f o r R n . I
A lth o u g h m an y d is ta n c e s em p lo y ed in m a th e m a tic a l o r sc ie n tific s e ttin g s a re m et rics. th e follow ing e x a m p le s p o in t o u t t h a t no n e o f th e ax io m s M l - M4 n e e d b e considered
j essential if d : X x X — ► [0. oc) is to d e s c rib e som e n o tio n o f d is ta n c e .
9
s
f E xam p le 3 .2 .6 S u p p o se .4 a n d B are c itie s connected by o n e -w a y rail lin es. The rail line fro m .4 to B is fiv e m ile s long, but the rail line fro m B to .4 is o n ly f o u r m iles long. Lid. X = {.4. B } a n d d efin e d : .Y x X —> [0. oc) by
d ( A . .4) = d ( B . B ) = 0. d ( A . B) = 5. d ( B . .4) = 4.
Observe that although d describes the rail dista n ces between cities, it does n o t sa tisfy axiom
M3. I
E xam p le 3 .2 .7 L et e > 0 a n d suppose th a t a real n u m b e r x m a y be u sed as an approxi m a tio n f o r a real n u m b e r g provided th a t the E uclidean d ista n ce fr o m x to g is less than s. IVe can view £ as the s m a lle st u n it m easurable by a p a rtic u la r m e a su r in g device. Then d* : R x R —> [0. oc) d efin ed by
[ 0. i f |ar - g\ < £: d €( x . g ) = <
I 1. otherw ise.
models this situ a tio n in th e se n se th a t it id en tifie s those real n u m b e rs "close e n o u g h " to a given real n u m b e r x to be considered a p p ro xim a tio n s f o r x . N o te th a t <L violates M 2 and
M4. |
E xam p le 3 .2 .8 L et Q be the se t o f ra tio n a l num bers in [0. I] a n d P - [0. 1] — Q . Given x € [0 . 1]. recall th a t x has a un iq u e decim a l represen ta tio n w hich does no t term in a te.
u
For each x G [0. 1] a n d each positive integer k . let x ( k ) he the k -th decim al digit in th is representation o f x . N o w define d : [0. lj x [0. I] —> [0. x :) by
i
d ( x . y ) = <
1. t / x . t / G P :
0. i f x . y G Q a n d x = y:
rjTr. otherw ise, where n is the least o f
all k G Z ~ such th a t x ( k ) ^ y( k).
The m o tiv a tio n behind the definition o f d ste m s fr o m the a tte m p t to appro xim a te an ir rational in [0. 1] usin g rationals in [0. lj. Sin ce it w ould be im practical to use irra tio n a ls as app ro xim a tio n s, we have constructed d so that irra tio n a ls become ' f a r " fro m one a n o th er (even fr o m th em se lv es). T hus, d does not s a tis fy M l . It should also be. noted that d does not s a tisfy M2 (d{ r r - r^-) = 1. since is irra tio n a l, and. as r r = 0 .7 0 7 1 0 6 ... .
* c / ( ^ . 0 . 7 l ) = d( 0.71. so that. d ( # > d ( ^ . 0 . 7 l ) + < /(0 .7 l. * f ) ) . O f course, in
a very concrete a n d u sefu l way. d does describe a n o tio n o f "d ista n c e " on the p o in ts o f[ 0 . lj
(points o f [0. 1] which are "close " to an irrational p G [0. 1], as measured by d. are ex a ctly
those rationals in [0 . 1] w hich "agree" with p in the f i r s t several decim al places). |
Q uite often, the m athem atical struc ture im posed (naturally or otherwise) oil a set can be represented in the form of a binary relation o n t h a t set.
E x a m p l e 3 .2 .9 For each n G Z ~ . the set {1 n } is called an initial segment o f 27*.
D efine
B f = {0} U {x|x : .4 —> {0. 1} f o r so m e in itia l segment. .4 o f Z ~}.
B,
=
{x|x : Z *-+
{0.1|}.
B f and B t are iso m o rp h ic to the sets o f fin ite a n d in fin ite bit strings, respectively. T hus.
B = B j U B t is the se t o f all bit strings. G iven x . y G B . we say that x is a prefix o f y.
i den o ted x C y . provided that the d o m a in o f x is a su b se t o f the dom ain o f y a n d f o r each k in the d o m a in o f x . x ( k ) = y{k). N ow define d r : B x B —► [0. oc) by
{
0 . i f x C i j :1. o th erw ise.
T hen, given x . y G B. x is a prefix o f y i f and only i f d r ( x . y ) = 0. Thus, d r is a rep re se n
O b se rv e t h a t C in tro d u c e d in E x am p le 3.2.9 is a b in a ry re la tio n on th e set o f all bit strin g s. T h e s itu a tio n d e sc rib e d in t h a t e x a m p le c a n b e g en e ralize d as follows. G iven a b in a ry re la tio n R o n a n o n e m p ty set X . define d y : X x X —> [0. oc) so th a t
{
(). if (x . y) € R: L. o th e rw ise .T h e n d y e n c a p s u la te s th e in fo rm atio n in clu d ed in t h e b in a ry re la tio n R in th e sen se that, (x. y) € R if a n d o n ly if d y ( x . y ) = 0.
F rom th e e x a m p le s considered, we have sh o w n t h a t w hile basic m etric ax io m s a re satisfied by m a n y c o m m o n d ista n c e fu n ctio n s, th e re e x is t useful m a th e m a tic a l stru c tu re 's w hich a re b e t te r re p re s e n te d by d ista n c e fu n c tio n s s a tis fy in g n o n e o r only som e o f those' m etric ax io m s. In th e lite ra tu re , th e d ista n c e fu n c tio n d efin e d in D efinition 3.2. L is sm n r- tim es re ferred as w eak dista n ce fu n c tio n ( wdf ) to e m p h a s iz e th e fact th a t it m ig h t not s a tisfy an y m e tric ax io m s [59].
B ased o n D e fin itio n 3.2.1. a d ista n c e fu n c tio n d on a set X has to be? defined for all (p. q) € X x X . H ow ever, for m any d o m ain s, su c h a “com prehensive" distance; m ig h t not b e n ec essary o r m ig h t n o t even exist.
D e f i n i t i o n 3 .2 .1 0 ( P a r t i a l D i s t a n c e ) G iven a .set X . a p a r tia l d istan ce em .V i.s a d is
tance fu n c tio n d defined f o r all (p . q ) € X \ x X y . where X_.\ a n d X y are n o n em p ty subnets o f X .
T h e re a re also tim e s w hen we a re on ly in te re s te d in th e d istan ce s o rig in a tin g from one p a r tic u la r fixed p o in t.
D e f i n i t i o n 3 .2 .1 1 ( P o i n t D i s t a n c e ) G iven a se t .V a n d a p o in t p 6 X . a p o in t distan ce' on p is a dista n ce fu n c tio n defined on {p} x X .
C learly, a p o in t d is ta n c e is a p a r tia l d is ta n c e by d e fin itio n . T h ere art; time;s wht;n we w ant to e x a m in e a d is ta n c e o r a p a r tia l d is ta n c e d from a sin g le p o in t perspective;, say po in t p. T h e p o in t d is ta n c e in d u ced from d on p is re p re s e n te d by d |p.