ແບບຮຽນຄະນິດສາດ ມ1

[qfmu !_ I6[g7yj’7n0v’,5, c]t 0v’glAo-nj

7;k,leobddPj ;da[g,af c]t glAo-nj
{ ra[g9Apczoj |b’j g0Qk c]h;g9ktI6[fh;p
xkp0v’;q’;Po s]n g0a,D
{ c[vvd gIqk9twfh glAo-nj XIv[ra[?
c]t lv’g,af M c]t N XIvpg9kt?
g[y’j I6[fhko0hk’D

!D dyf9tde

!D lhk’g,afg7y’j 7n0v’g,af E  H
dD

{ .-h8kdk3i0v’xN,0Pogrnjvlhk’I6[fhko F I 

0hk’oU7noF \kpg,af E, F, G, H, I, J. ສະຫງສ.ວວນ.ລິສຂະສິດ

9q’j .-h[afmaflkd grnjv\kpg,af A 16 [EG] D G d  J
{ .-h[aomaflkd lhk’g,af B g7yj’7nda[g,af

A mP[.ljcdo dD 9q’j vtmy[kp;jk ,ulv’;ymu9af;k’D

0D

{ .-h8kdk3i0v’xN,0Pogrnvj lhk’ 1  M  d  R 
N  S 
I6[fhko0hk’oU7noF \kpg,af M,N, 6 
6 
R c]t S, 9kdoAo .-h;q’;Pogrnjv O 
\kpg,af O D

{ .-h;q’;Po lhk’g,af T g7yj’7nda[

g,af O mP[.lcj do d rhv,ma’[vd;ymulhk’D

@D lhk’glAog7y’j 7n0v’glAo-nj

8q;1kj ’G .oc8j]td=]tou 8+wxoUF lhk’glAo-njg7yj’7n0v’glAo-nj k mP[.ljglAo-nj dD

a. k b. k

dd

147 

a. B k \kpg,af A c]t B .lj k c]h;lhk’g,af
A A' c]t B' g7yj’7nda[g,af A c]t B mP[

d .lj dD 0ufglAo-nzj kj o A' c]t B' gIqkwfh

(A'B') c,joglAo-njg7y’j 7n0v’ k muj8hv’lhk’D

A' B' gIqkla’gdfgsao;jk d c]t k 8afdao16gj ,af
b.

I Ak I D faj’oAo ;k’g,af A 16j k c]h;lhk’g,af
d A' g7yj’7nda[ A mP[.ljglAo-nj d. 0ufglAo-nj
zkj o I c]t A' wfhglAo-nj (IA') c,joglAo

A' -ngj 7y’j 7n0v’ k mu8j hv’lhk’D

#D lhk’I6[g7y’j 0v’,5,

{ oe.-h[aomaf c]t [aomaflkd grnvj -Uc9’;jk
A d g,af A c]t F; B c]t E 0v’I6[fhko

0hk’c,jog7yj’7ndaomP[.lgj lAo-nj dD
ສະຫງສ.ວວນ.ລິສຂະສິດ
B { AnBC = 45D S 3fp[+.-h[afmafcmd,5,F EnFC = ?
A' { AB = 3cm S 3fp[+.sh.-h[aomaf,u0uf\kpF

C [EF] = ?

B'

@D .97;k,

!D dkog7y’j 7nmP[.ljglAo-n|j b’j

dD glAodk’lkd0v’mjvo-n|j bj’ c,oj glAo-nGj A
{ zkj og,afg7y’j dk’0v’mvj o-nSj

{ 8A’lkd.shmjvo-nfj a’j djk;D g,afg7yj’dk’0v’

[AB] B

148 

0D g,afg7yj’7n0v’g,af A mP[.lgj lAo-nj ΔD

d=]toug,af A 1j6glAo-nj ΔD d=]toug,af [+1j6glAo-nj ΔD

X A∈ Δ ? X A∉ Δ ?

cdog7y’j 7n AA

A Δ Δ A'
Δ
g,afg7yj’7n0v’g,af A mP[.lgj lAo-nj Δ
g,afg7yj’7n0v’g,af A mP[.lj c,jog,af A' g-yj’glAo-nuj Δc,oj glAodk’
glAo-nj Δ c,jog,af A oAogv’ lkd 0v’mjvo-nj [ AB ]D
X 8q;,aogv’ ? X A c]t A' g7yj’7ndaomP[.ljglAo-nj Δ?

@D rNo4ko0v’dkog7y’j 7n c]t 75o]adltot ສະຫງສ.ວວນ.ລິສຂະສິດ
8q;1jk’G

g,afg7yj’7n0v’[aofkg,afmuj]Poc4;fP;dao d=c,oj [aofkg,af]Poc4;daoS glAo-nmj uj
g7yj’7nda[glAo-n|j bj’d=c,joglAo-nDj
I6[g7yj’7nG

{ 0v’mvj o-n|j bj’ c,jomvj o-nmj u]j ;’pk;gmqkj daoS
{ 0v’,5,|bj’ c,oj ,5,muj,u7kj ;afcmdgmqkj daoS
{ 0v’;q’,qo c,jo;q’,qomu,j u]aflt\ugmqkj daoD

149 

dD lv’I6[g7y’j 7ndao

lv’I6[g7yj’7ndao c,oj lv’I6[gmqkj daoF lv’mjvo-ngj 7yj’7ndao c,jor;d,ao,u];’

pk;gmqkj daoF lv’,5,g7yj’7ndao c,jo7jk;afcmdgmqjkdaoF lv’;q’,qog7y’j 7ndaoc,jo

,u]aflt\ugmqjkdaoD C  C   d 
8q;1kj ’G
E   C ' 

A  D 
r   r' 

F  B 

mjvo-nj [AB] c]t [DF] g7yj’7ndaomP[.lj d gIqkwfh AB=DF
,5, FnAB c]t BnDF g7y’j 7ndaomP[.lj d gIqkwfh FnAB = BnDF

I6[;q’,qo C c]t C ′ g7yj’7ndaomP[.lj d gIqkwfh r = r’D

0D I6[g]0k7toyf ສະຫງສ.ວວນ.ລິສຂະສິດ
 I6[lk,c9mPj ’ cdog7yj’7n
,5,rNo
{ I6[lk,c9mjP’,u|b’j cdog7yj’7nG
glAodk’lkd0v’rNoD

{ ,5,0v’rNo0v’I6[lk,c9mPj ’
c,jogmqjkdaoD

 I6[lk,c9dltg\u

I6[lk,c9dltg\u,ulk,cdog7yj’7nmuzj kj og,affP;daoD
{ I6[lk,c9ltg\u c,oj d=]touryglf0v’I6[lk,c9
mPj ’D
{ m5d,5,0v’I6[lk,c9ltg\u ,u7jkgmqjkdao c]t gmqkj
da[ 60D D

150 

 I6[fvd9ao ສະຫງສ.ວວນ.ລິສຂະສິດ
{ ,ulv’cdog7y’j 7n8A’lkddaoS
{ glAogoa’j 9v,8A’lkddao c]t
,ug,afg7yj’dk’fP;daoD

 I6[lujc9lkd
{ ,ulv’cdog7’jy 7n g-yj’c,joglAodk’lkd
0v’0hk’D
{ glAogoa’j 9v,,ug,afg7y’j dk’fP;dao c]t
,u];’pkdgmqkj daoD

 I6[9t85]af
{ I6[9t85]afgxaoma’I6[fvd9ao c]t
gxaoma’I6[lucj 9lkdD
{ ,ulucj dog7yj’7nD

{ glAogoaj’9v,,ug,afg7y’j dk’vaofP;daoS
,u];’pk;gmqjkdao c]t 8A’lkddaoD

151 

8q;1jk’G lhk’g,afg7y’j 7n0v’g,af M mP[.lgj lAo-nj d fh;p;q’;PoD

I6[ dD I6[ 0D M  

M  A  B  d 

A  B  d 

I6[ dD \kplv’g,af A c]t B .lgj lAo-nj dD M'  

I6[ 0D 0ufmjvodqj’0v’;q’,qomu,j ug,af.9dk’ A g-yj’zkj og,af M c]t 0ufmjvodqj’

muj,ug,af.9dk’ B g-yj’zkj og,af MD lv’mvj odqj’8afdao16jg,af

M' g-y’j gxaog,afg7y’j 7n0v’g,af M mP[.lmj vj o-nj AB X s^n glAo-nj d ?D

#D [qfg/ydsafD ສະຫງສ.ວວນ.ລິສຂະສິດ

!D J  F 

.odkog7yj’7nmP[.ljglAo-nj d F J c,jog,af

I   K   g7y’j 7n0v’g,af IF K c,oj g,afg7yj’7n0v’

g,af HF F c,oj g,afg7yj’0v’g,af ED

E 

H   d   dD mjvo-n.j fgxaomvj og7yj’7n0v’mvj o-nj
[HI] L c]t mvj o-nj.f gxaomvj o-njg7yj’7n0v’ [HJ] L

0D E c,jog,afg7y’j dk’0v’mvj o-nj [HI] le]a[g,af F lk,kflts]5[wfhco;.f L

gxaospa’ L

@D 9q’j c8h,I6[0hk’0;koU7no 3fp;jk A  N  P 

AM=3cm; MN=4cm; NP=2cm M 
Δ 
c]t d//(NA) 9kdoAoF 9q’j oe.-h
A'  
[aomaf c]t [aomaflkd grnjvc8h, d 
glAo-nj d' mujg7yj’7nda[glAo-nj d mP[.ljglAo-nj Δ D

152 

#D .oI6[8+wxoUG I  A 
{ (AI) ⊥ (AJ) B  J 
{ B c,oj g,afg7y’j 7n0v’ A mP[.lj
glAo-nj XIJ?D

dD .odkog7yj’7nmP[.ljglAo-nj XIJ?F
c,oj glAo-nj.fg7y’j 7nda[glAo-nj XAI?L
c]t glAo-n.j fg7yj’7nda[glAo-nj XAJ?L
l;j oglAo-nj XBI? c]t XBJ?9tlts]5[;jkco;.fL

0D 9q’j dkj p7noxt3spd8+wxoU c]t 8nj,.l[j vj o9Eg,afG

< 4hk;jklv’glAo-nsj kd8A’lkddao c]h; I6[mugj 7yj’7n0v’r;d,ao

8A’lkddao DD D D D<

$D dD 9qj’c8h,I6[lk,c9mjP’ PIC muj,u9v,1j6g,af I c]t 7kj ;afcmd PnIC = 55D c]t

ສະຫງສ.ວວນ.ລິສຂະສິດ
IP = 6cm

0D 9qj’0ufmjvo-nj [xy]. 4k,;jk9t;k’g,af I w;h[jvo.f c]t 16gj lAo-nj.f grnjv.shg,af I

gxao9v,0v’I6[ # c9mPj ’ xnTy D C  A 
7D 9q’j c8h,glAo-nj dF \kpg,af A, B, C, D 8k,

I6[mu1j 6j0hk’0;koUD 9kdoAo 9qj’\kpg,af C', D' D 
mugj xaog,afg7y’j 7n 0v’g,af C c]t D 8k,]efa[ B 
mP[.lgj lAo-nj d. 9qj’[vd[aofkI6[lk,c9m5dMI6[ d 
mujwfhD

&D .oI6[0hk’0;koUG C  I  C ′ 
{ O c]t O’ g7yj’7ndaomP[.lgj lAo-nj dD G 
{ I c,jog,af|b’j 0v’glAo-nj dD F 
{ C c,oj ;q’,qomu,j u.9dk’ O c]t ,u O  O'  

]aflt\u !F@cmF C ′ c,jo;q’,qomuj,u d 
mu,j u.9dk’ O c]t ]aflt\u !F@ cmD
dD .odkog7jy’7nmP[.ljglAo-nj dS
{ g,af.fg7y’j 7nda[g,af IL

153 

{ mvj o-n.j fg7yj’7nda[mvj o-nj [OI]L
{ ;q’,qo.fg7y’j 7nda[;q’,qo C L
0D mjvo-nj [OI] 8af;q’,qo C 1j6g,af FS mjvo-nj [O’I] 8afda[ C ′1j6g,af GD
9qj’[vdg,afmujg7yj’7nda[ g,af F mP[.lgj lAo-nj dL
{ 9qj’dkj p7noxt3spd8+wxoU c]h;0Po8n,j .l[j vj o9eg,af.sh4nd8hv’D

F 16jmvj o-nj [OI]F g,afg7y’j 7n0v’,aoc,jog,afmuj16jmvj o-nj D D D F g7yj’7nda[ [OI]
F 16j;q’,qoD DD D D F g,afg7y’j 7n0v’,aog-yj’c,jog,afmuj1j6;q’,qoD D D DDF
g,afg7y’j 7n0v’g,af F c,jog,af D D D D D 

*D .oc8]j td=]tou8+wxoUG a. A d
{ 0ufglAo-nj d .ljg9Ap0k;D b. B
{ \kplv’g,af A c]t B F 9kdoU0uf
glAo-nj XAB? c]t lhk’I6[g7yj’7n0v’ I
,aomP[.ljglAo-nj dD
ສະຫງສ.ວວນ.ລິສຂະສິດ
AB
d

c. A
d

(D I6[lk,c9g7yj’7n B

I C5 dD .-hg9Ap0k;D
2 { lhk’I6[lk,c9 ABC 3fp;jkG
B 10 J AB=2cm, AC=9cm, BC=10cmD
d { 0ufglAo-nj d 8afda[glAo-nj XAB? 1j6g,af I
6 c]t 8afglAo-nj XAB? 1j6g,af J fa’j I6[0hk’
9 -hkpoUD

A

154 

0D .-hj;q’;Polhk’g,af A′, B′, C′ g7y’j 7nda[g,af A, B, C mP[.ljglAo-nj dD
0ufI6[lk,c9 A′B′C′D
7ecotoeG .shoe.-hg,af I c]t J 0v’glAo-nj dD

)D oe.-h[aomaf,u0uf\kp c]t [aomafcmd,5,grnvj G 4 cm  
dD ]vdc[[I6[muj1j60hk’0;koU8k,0t|kf8q;9y’.lj 40D  

g9Ap0k;D d 
0D lhk’g,af B g7y’j 7n.shg,af A mP[.lgj lAo-nj dD

_D lhk’I6[fhko0hk’oU7no8k,0t\k,8q;9y’g,af
.lgj 9Ap0k;0ufglAo-nj d 7nda[38mujltcf’.oI6[F
9kdoAo lhk’I6[mujg7yj’7n.sh,5,mu,j u1j60hk’oU3fp
mP[.ljglAo-nj d D

ສະຫງສ.ວວນ.ລິສຂະສິດ

155 

[qfmu @W cdog7yj’7n0v’I6[F glAodk’lkdF glAoc[j’g7y’j ,5,

7;k,leobd djP;da[cdog7yj’7n0v’I6[D I6[lv’vao1j6]j5,oUwfhlhk’0Bo fh;pc,cj [[ μ
vaofP;daoF μ 4nd]vd8q;c[[lv’gmnvj 1j6.oc8j]tI6[D g,af A dq’daoda[g,af A′ D

A 1 2 A' 1 2 

A

I6[mu ! c,jc[[ μ I6[mu @ A'

.og,njvra[czjog9Ap.sh A wxg8a’da[A′S .og,njvra[czog9Apgrnjv.sh A wfhg8a’da[A′S
l;j omu X!? c]t X@? 0v’I6[ ! g8a’dao l;j omu X!? c]t lj;omu X@? 0v’I6[ @
r=fuD [+g8a’daoD
glAo-nj d 0v’Ivpra[c,oj cdog7y’j 7n I6[ @ [+,ucdog7y’j 7nD
0v’I6[ !D
ສະຫງສ.ວວນ.ລິສຂະສິດ
A
A'

A
d d A'

1D dyf9tde

!D cdog7yj’7n
dD .oc8j]td=]touF .shoe.-hg9hp0k;F [aomaf,u0uf\kp c]t [aomaflk,c9lkdF 9kd
oAod;fg[y’j fh;pdkora[D
{ 0ufglAo-nj dF \kpg,af M 3fp;jk M∉dD gfqk X dt ? ;jkg,nvj gryjora[ c]t gryoj g9kt
I61j6g,af M g\nvoda[0= Xd?G lhk’ N g,af8yf8+daoD
{ \kpg,af M c]t N 3fp;kj MN = 8cm D Ivd7noIvpra[oAoD
0D 8afg9hp]vdc[[.shgxaoI6[9t85]af,u0hk’ $ cm .shwfh @ czjoD

czomu|b’j .sh]vdc[[I6[F! c]t glAo-nj d1D ra[g9hp]vdc[[8k,glAo-nj d1D
la’gdfgsaospa’L d1 c,oj cdog7y’j 7n0v’ F! [+L

156

ສະຫງສ.ວວນ.ລິສຂະສິດg-ajofP;daoda[0hk’gmy’oUF le]a[I6[ F@
c]t glAo-nj d2D 4k,;kj I6[F@ ,ucdo
g7yj’7n[+L

@D glAodk’lkd0v’mjvo-nj
dD oe.-hg9hpdkogd 0ufmjvo-nj [AB] muj,u];’pk; 6cm S \kpg,afg7y’j dk’ ID
{ .-h[aomaflk,c9lkdF 0ufglAo-nj d 8A’lkd.shda[mjvo-n1j 6jg,af ID d gvUo;kj

glAodk’lkd0v’mjvo-nj [AB]D
{ fh;pdkora[g9ApdkogdF d;fg[y’j ;kj

d c,oj cdog7yj’7n0v’mvj o-nj [AB] [+ L
{ \kpg,af M .ljglAo dD 9qj’dkj p7no c]h;
8nj,.l[j vj o9Eg,af.sh4nd8hv’G

cdog7yj’7n0v’mjvo-nj [MA] .od=]toumP[
.lj DDD c,oj mvj o-njD D D F g-y’j MA=. . . ..

gIqkg;Qk;jkg,af M sjk’ltg\u9kdg,af A c]t g,af BD
{ 3f[oe.-h;q’,qoF lhk’g,af N, P, R 3fp;kj G NA=NB=5cm, PA=PB=5cm,

RA=RB=7cm, SA=AB=7cmD [aofkg,af N, P, R, S 9t1j6glAo-nsj pa’ L
0D dkolhk’c8h,glAomk’lkdfh;p;q’;PoD

oe.-hg9hp0k;F 0ufmjvo-nj [AB] mu,j u];,pk; *cmD vhk0k;q’;Po.shwfh7;k,
d;hk’ (cm \kpg,af M, P sjk’ltg\u9kd A c]t 9kd BD
0ufglAo-nj XMP?D d;f7no;jk,ao8af c]t 8A’lkd16jg7y’j dk’0v’mvj o-nj [AB] [+ L
glAo-njfa’j djk;gxaospa’0v’mjvo-nj [AB] L

157

#D glAoc[j’g7yj’,5,0v’,5,|b’j
dD .-hg9hpw0,aoF 0ufg7yj’glAo-nlj v’glAo [ox) c]t [oy )D ra[g9hp3fp.sh [ox) c]t
[oy ) g8a’dao 9kdoAoc[g9hpvvdF 0ufglAo-nj d 8k,Ivpra[D

x  x 
y

O  y O 

0D glAo-nj d c,jocdoj g7y’j 7n0v’I6[mujxtdv[fh;pg7y’j glAo-njlv’glAo [ox) c]t [oy )D
I c,jog,af0v’glAo-nj dF gIqkgvUo ,5, xmoI c]t Imoy ;kj DDDDDDDDDDDDDDDDD
9q’j xP[mP[7kj 0v’,5,faj’dkj ;D

ສະຫງສ.ວວນ.ລິສຂະສິດ
2D .97;k,

!D cdog7y’j 7n0v’I6[

glAo-nj d c,oj cdog7yj’7n0v’I6[F gIqkwfh I6[g7y’j 7n0v’ F mP[.lj d c,oj I6[ F
oAogv’D
8q;1jk’G glAo-nj d c,oj cdog7y’j 7n0v’I6[D gryjolk,kfd;f7nofh;p
g9hp]vdc[[I6[c]h;ra[D

d  d 
158

@D glAodk’lkd0v’mjvo-nj
glAodk’lkd0v’mvj o-nj|b’j c,oj glAo-nmj uj8af c]t 8A’lkd 16jg,afg7yj’dk’0v’
mjvo-njoAoD

M  M 

A  B  A  B 
Δ  Δ 

M c,joglAodk’lkd0v’ [AB] AM = BM

75o]adltotG

{ 4hk;kj g,af|b’j 1j6glAodk’lkdF .o0totoAog,affa’j djk; c,oj sjk’ltg\u9kdlv’

lQo0v’mvj o-nDj ສະຫງສ.ວວນ.ລິສຂະສິດ

{ 4hk;jkg,af|b’j sjk’ltg\u9kdlv’lQo0v’mjvo-nj|bj’F g,afoAo 1j6glAodk’lkd0v’

mjvo-njfaj’djk;D

 dkolhk’c8h,fh;p;q’;PoD

g8addoyd ! g8addoyd @

B
A

A

B

dkoc8h,glAodk’lkd0v’mjvo-nj|b’j F lk,kflhk’fh;plv’g,afmujsjk’ltg\u9kd
lv’lQo0v’0v’mjvo-njoAoD

159

#D glAoc[j’g7yj’,5,0v’,5,|bj’glAo x

c[’j g7yj’,5,0v’,5,|bj’c,joglAo-nj glAoc[j’g7y’j ,5,0v’,5, xnOy

mucj [’j ,5,oAovvdgxaolv’,5,gmqkj daoD O d
y
BnAx = xnAC = 1 BnAC
2
\kpgsfG glAoc[’j g7yj’,5,0v’,5,|b’j

c,oj cdog7yj’7n0v’,5,oAoD

7ecotoe.odkolhk’c8h,
dD lhk’c8h,glAodk’lkd0v’mjvo-njD
8q;1jk’G .-hg9hp0k;F 0ufmvj o-nj [AB]F .-h[aomaf c]t ;q’;PoF lhk’c8h,glAodk’
lkd0v’mjvo-nDj

!D @D

II

A BA B

JJ
ສະຫງສ.ວວນ.ລິສຂະສິດ

{ 0uflv’mjvodqj’mu,j ug,af.9dk’16j A c]t B fh;p]aflt\us^kpd;kj s^n gmqkj da[ [AB]

Xma’lv’mjvodq’j 8hv’,u]aflt\ugmqkj dao?D lv’mjvodq’j 8afdao16jg,af I c]t JD

{ 0ufglAo-nzj jko I c]t J gIqkwfh glAodk’lkd XIJ? 0v’mjvo-nj [AB].

0D lhk’c8h,glAoc[’j g7y’j ,5,0v’,5,|bj’D

8q;1jk’G .-hg9hp0k;F 0ufg7y’j glAo-njlv’glAo [ox) c]t [oy)D

oe.-h[aomaf c]t ;q’;Polhk’c8h,glAoc[’j g7y’j ,5,0v’,5,

!D @D

y y

J J rM
OI
x Or
Ix

160

{ 0ufmvj odq’j muj,u.9dk’ O c]t lt\u r muj8afdao [ox) 16j I c]t [oy) 16j JD
{ 0uflv’mjvodqj’mugj ,af.9dk’16j I c]t J fh;p]aflt r vaofP;daoD
{ lv’mvj odqj’8afdao16jg,af MD
{ 0ufglAo-njzkj og,af O c]t M. gIqkwfh glAo-nj XOM? c,joglAoc[j’g7yj’,5,0v’,5,

xnoy D

#D [qf/bdsaf

!D I6[1j60hk’0;koU ,ucdog7yj’7n[=L

4hk,uF .sh]vdc[[I6[oU7no c]t

0ufcdog7yj’7nD4hk[+,uF .shfafcdh

I6[oU grnjv.shI6[,ucdog7yj’7nD

@D dD .sh]vdc[[I6[oU7no3fp.-h8kdk3i

0v’xN,0PoD 1.  2.  3. 
5. 
0D -vdsk;jkI6[faj’djk;F ,u|b’j Xs^ns^kp? ສະຫງສ.ວວນ.ລິສຂະສິດ

cdog7yj’7nD 4hk,uF 9qj’0ufcdofa’j dkj ;

fh;pglAolucf’D

4. 

#D 9q’j \kp9eo;ocdog7y’j 7nle]a[xhkplao
pko9k]t9vo0v’I6[]j5,

$D dD .-hczog9Apmu,j u8kdk3i 0ufmjvo-nj [AB] 3fp;kj AB=10cmD \kpg,af C
.ljmjvo-njfaj’dkj ;3fp AC=6cmD

161

ສະຫງສ.ວວນ.ລິສຂະສິດ0D oe.-h[aomaf c]t [aomaflkd lhk’glAodk’lkd0v’ [AB] c]t [AC]D
&D 9qj’c8h,mvj o-nj [AB] c]t glAodk’lkd0v’,ao \kpg,af I .ljglAodk’lkdoAo

vud3fp.shglAo AB gxaoglAodk’lkd0v’mvj o-nj [IH]D
*D 9q’j c8h,I6[lk,c9 ABC. .lj;q’;Poc8h,glAodk’lkd0v’0hk’ AB, AC, BC

gIqkla’gdfgsaoco;.fL
(D .shI6[lucj 90hk’0t|ko ABCDF glAo-nj d! c]t d@ gxaoglAodk’lkd0v’mvj o-nj

[AB] c]t [CD]D 9qj’xP[mP[mu8j A’0v’lv’glAo-nfj aj’dkj ;D
)D .shlk,g,af A, B, C [+1j6glAo-njfP;daoD 9qj’c8h,glAodk’lkd0v’mvj o-nj [AB]

c]t [AC]D lv’glAodk’lkd8afdao1j6g,af ID 9qj’-Uc9’;kj G I 16jglAodk’lkd
0v’mjvo-nj [BC]D 4hk;kj A, B, C 1j6glAo-nfj P;dao9tgxaoco;.fL
_D lhk’c8h,glAoc[’j g7y’j ,5,0v’,5,|bj’D
dD lhk’,5, xnoy 3fp;kj xnoy = 70D
0D .-h[aomaf c]t ;q’;Po lhk’glAoc[’j g7yj’,5, d 0v’,5,faj’dkj ;D
7D .-h[aomafcmd,5, d;f7no;kj d c[j’ xnoy gxaolv’,5,mu,j u7kj ;afcmd 35D D
!WD dD lhk’I6[lk,c9 ABC 3fp;jk BC=10cm, AB=6cm, AC=9cmD
0D .-h[aomaf c]t ;q’;Po lhk’glAoc[’j g7y’j ,5,0v’,5, BnAC glAofaj’dkj ;8afmvj o
-nj [BC] 16jg,af ID 9q’j ;afcmd BI c]t ICD g,af I gxaog,afg7yj’dk’0v’mjvo-nj
[BC] [+L
7ecotoe G le]a[g,af.shg[y’j 7ecotoe.odkolhk’c8h,glAoc[’j g7yj’,5,0v’,5,

|b’j zkj o,kD
!!D dD lhk’c8h,I6[lk,c9 ABC 3fp;jkG BnAC = 110D , AB=6cm, AC=10cmD
0D oe.-h[aomaf c]t ;q’;PoF lhk’glAo-nj d mujgxaoglAoc[’j g7yj’,5, BnAC ,ao8af

mvj o-nj [BC] 16jg,af ID 9qj’7yfw]j7kj ;afcmd0v’ BnAC D
7D oe.-h[aomaf,u0uf\kpgmqjkoAoF \kpG

{ g,af E g,afg7y’j 7n0v’ B mP[.lgj lAo-nj dD
{ g,af F’F g,afg7y’j 7n C mP[.ljglAo-nj dD
c8h,I6[lk,c9 AEF g-y’j ,ao8afda[mvj o-nj [BC] c]t [EF] [+L lhk’c8h,
cdog7y’j 7n0v’r;d,aoD

162

!@D I6[8+wxoU AB c,oj mvj odq’j 0v’;q’,qo,u.9dk’ OD

dD 9q’j ]vdI6[oU7no.lgj 9hp0k; 3fpgvqk AD=4cmD

0D oe.-h[aomaf c]t ;q’;Po lhk’c8h,cdog7y’j 7n A 130D B

0v’I6[faj’dkj ;F cdog7yj’7nfa’j dkj ;,udkorq;rao

da[,5, AnOB co;.f L

!#D I6[8+wxoUwfhlhk’0Bofh;plv’mjvo-nj [AB] c]t A C
[CD] muj,u];’pk;gmqjkda[ c]t gmqjk &cm c]t
B
8afdao16jg,af I 3fp;jk AI=IC=2cmD C D

dD 9qj’]vdI6[oU7no.ljg9hp0k;3fpgvqk DnIB = 190D D

0D oe.-h[aomaf c]t ;q’;PoD 9q’j lhk’c8h,cdog7y’j 7n0v’I6[faj’djk;gryjoIhv’

I6[lucj 9 ACBD ;kj co;.fL

!$D 9q’j la’gdf c]t ;yw9D C A
dD d;f7nofh;p[aomaf c]t [aomaflkdg-yj’glAo-nj

[AB] c]t [CD]D 8A’lkddao c]t glAo-nj XCD? AC = CB
AD = DB ສະຫງສ.ວວນ.ລິສຂະສິດ
8af [AB] 16jg,afg7y’j dk’D
B
0D 9qj’dkj p7no c]h;xtdv[xt3spd]5j,oU.shlq,[6oD

4hk|bj’g,af,uw]ptsjk’ltg\u.shlv’lQo0v’mvj o-nFj 9twfh D D D D DD D D DDD D D DD DD D DD D

.og,nvj AC=CB, C c,jo D D D DD D D D D D D D D DDD DD DD D D D D

.og,njv DD D DD DD D DDD D DDD DD DDF D c,jo D DD D D D D D D D D DDD DD DD D D D

glAodk’lkd0v’mjvo-nj [AB] c,oj DDD D D D D D D D D D D DD D D DDD D

s]n glAodk’lkd0v’mjvo-nj|b’j 8af{8A’lkd.sh,ao16jg,afg7yj’dk’F

faj’oAo XAB?D D D D D DDXCD? c]t XCD?D D D DDD DD DD D

!&D d D oe.-hg9hp0k;F c8h,mjvo-nj [AB] 3fp;jk AB=10cm. \kpg,af C .ljmjvo
-nj3fp;kj AC=6cmD

0D oe.-h[aomaf c]t [aomaflkdlhk’c8h,glAodk’lkd0v’mvj o-nj [AB] c]t glAo
dk’lkd0v’mjvo-nj [AC]D 8+da[glAodk’lkdma’lv’oU gryoj lts^5[wfhco;.f L

163

!*D dD oe.-hg9hp0k;F 0uf;q’,qomuj,u.9dk’ O mu,j u]aflt\u 4 cm. ;k’g,af A

c]t g,af B .lj;q’,qoD

0D oe.-h[aomaf c]t ;q’;Polhk’glAodk’lkd0v’mvj o-nj [AB]D

gxaos[a’glAodk’lkd9bj’zjkog,af.9dk’ O L d
!(D [vj ozyf1j6 ( [vj o16jI6[8+wxoU grnvj .shI6[0hk’

0;klk,kfgxao8q;g7y’j 7n 0v’I6[0hk’-hkp

,nmP[.ljglAo-nj dD

!)D dD c8h,;q’,qo C muj,u.9dk’ O c]t A B d
]aflt\u 4cm c]t glAo-nj d g-y’j 8af
.shglAo C D C J

0D \kpg,af A, B, C 3fp;jkG O ສະຫງສ.ວວນ.ລິສຂະສິດ
{ XOA? 8af.shglAo-nj dF
{ XOB? 8af.shglAo-nj dF C
I

K

{ XOC? 8af.shglAo-nj dF

g,af I, J, K c,oj g,af8af8k,]efa[1j6glAo-nj d faj’ltcf’.oI6[c8h,0hk’gmy’oUD

7D 9q’j lhk’;q’,qo C’ ′ I6[g7y’j 7n0v’;q’,qo C mP[.lgj lAo-nj dD gIqkgvUo O′ ;jkG .9

dk’0v’ C ′D
’D .odkog7y’j 7nmP[.ljglAo-nj dF glAo-n.j fgxao8q;g7yj’7nda[glAo-nj XIO?F XJO? c]t

XKO?L 9qj’lhk’c8h,g,af A′, B′, C′ g7yj’7n A, B,C mP[.ljglAo-nj d 9kdoAo c8h,

I6[lk,c9 A′ B′C′D

!_D oe.-h[aomafc8h,D A
dD 0ufglAo-nj d c]t lhk’lv’g,af A c]t B d

g7y’j 7ndaomP[.ljglAo-nj dD C

0D \kpg,af C 3fp;jkglAo-nj XAC? c]t B

XBC? 8afglAo-nj dD lhk’g,afg7y’j 7n0v’g,af C mP[.ljglAo-nj d rhv,vtmy[kp;ymulhk’D

@WD I6[lk,c9lkdD

dD lhk’I6[lk,c9 ABC lkd1j6 A 3fp;jk AnCB = 60D c]t CA=3cm D

164

ສະຫງສ.ວວນ.ລິສຂະສິດ0D lhk’I6[lk,c9 BCE g7y’j 7nda[I6[lk,c9 ABC mP[.lgj lAo-nj XBC?D
I6[lk,c9 BCE gxaoI6[lk,c9xtgrf.f L

7D lhk’I6[lk,c9 FAC g7y’j 7nda[I6[lk,c9 ABC mP[.lgj lAo-nj XAC?D
’D la’gdfgsaog,af F, C, E gxaoco;.fL
9q’j dkj pxt3spf]j5,oU7noc]h;0Po8n,j .lgj rnjv.shxt3spd]5j,oUlq,[6oD

Ih6;kj ,5,g7y’j 7ndaomP[.ljglAo-nj|b’j ,u7jkgmqjkdaoD ,5, EnCB c]t BnCA g7yj’7ndao
mP[.lgj lAo-nj XBC? c,jogmqkj daoG D D DD DD D = . . . .. . = 60D
g-ajofP;daoF ,5, D D D D DD c]t D D D D D D mjug7yj’7ndaomP[.ljglAo-nj D D D DF c,jogmqjkdaoG
D D D D D DD = ………= 60D
s^nF EnCF = EnCB + BnCA + AnCF

EnCF = .. . . ..+ . ….. + .. . .. . .

,5, EnCF c,jogmqkj daoD D D D DDS g,af E, C c]t F c,jo D D D D D D D D D DDD D D

@!D dkog7y’j 7n0hk,cdoD
9q’j 8v[fh;p <zyf< s^n <4nd< .lj0h=mujgsao;jkg\ktlq,.ljD

!D 4hk;jklk,g,af]Podao16jglAo-njfP;daog,afg7yj’7n0v’r;d,aod=1j6glAo-nfj P;daoD
@D 4hk;jklujg,af1j6;q’,qofP;daoF g,afg7y’j 7n0v’r;d,aod=16j;q’,qofP;daoD
#D I6[lujc9 c]t I6[g7yj’7n0v’,ao,u];’Iv[gmqjkdaoD
$D mjvo-nj|b’j ,uc8|j bj’cdog7yj’7ngmqkj oAoD
&D [aofkglAo-nmj uzj jko.9dk’0v’;q’,qogxaocdog7yj’7n0v’;q’,qooAoD
*D glAo-nj|bj’,uc8jlv’cdog7yj’7ngmqjkoAoD
(D 4hk;jk MI=MJ \kp;kj M c,jog,afg7yj’dk’0v’mjvo-nj [IJ]D
)D ,aoxkdqf,ulv’g,afmuj1j6w]ptfP;da[g,af A c]t g,af BD
_D glAoc[’j g7yj’,5,0v’,5,lkdwfhc[j’,5,fa’j dkj ;vvdgxaolv’,5,mugj mqjkda[ 45D D
!WD glAoc[j’g7yj’,5,0v’,5,|bj’wfhc[’j ,5,fa’j djk;vvd

gxao,5ocs],lv’,5,mujgmqkj daoD

165

@!D dD 9q’j ]vdc[[I6[8+wxoU8k,0t|kfmu.j shD A 

0D lhk’g,af E c]t F g7yj’7nda[g,af B c]t C 40D  
mP[.lgj lAo-nj dD 6 cm   4 cm  

7D 7yfw]]j ;’Iv[0v’I6[lk,c9 AEFD B   3cm   C   d 
’D glaAo-ngj xaospa’0v’,5, CnAF L c]t gxao

spa’0v’mvj o-nj [BE]L

9D glAo-nj XBC? c]t XEF? 8afdao1j6.l L d’

@@D 1j6I6[0hk’0;koUF C! c]t C@ c,jo EA BF d
lv’;q’,qomu,j u.9dk’ O vaofP;daoS O

glAo-nj d 8af.sh C! 1j6g,af A c]t B C! C@
c]t 8af.sh C@ 16jg,af E c]t FD
glAo-nj d’ 8af8A’lkd.shglAo-nj d muzj jkog,af.9dk’ OD

dD .odkog7yj’7nmP[.ljglAo-nj d’F 38.fc,oj 8q;g7y’j 7nda[ C! L da[ C@ L da[ dD
ສະຫງສ.ວວນ.ລິສຂະສິດ
0D 9tlts^5[co;.f8+da[g,af A c]t g,af BL da[g,af E c]t g,af F L le]a[7;k,

pk; AE c]t ];’pk; BFL

@#D 9qj’lhk’I6[8+wxoU 8k,0t|kfmujdeoqf.sh c]h;0ufcdog7y’j 7n0v’r;d,aoD

1.  2.  A B
A AB 3.  a a
r 40D rc b
C! C@ b
r = 4 cm B c

rr12 C 2 cm a = 1cm r = 2 cm
= 3 cm b = 4 cm c = 2 cm
=

@$D 9q’j lhk’lv’mjvo-nj [AB] c]t [AC] 3fp;kj G AB=AC=5cm. BnAC =85D
9kdoAo lhk’glAodk’lkd c]t glAoc[j’g7y’j ,5,0v’,5, BnAC D

166

ສະຫງສ.ວວນ.ລິສຂະສິດ [qfmu @! ,kf8kl;j o

!D dyf9tde

m;o7nodkosp=h c]t 0tspkpI6[
dD 9q’j la’gdfg[y’j I6[-hk’ XI6[!? mujc8h,.oI6[lucj 9lkdmu,j u0t|kf 20×12

8kdk3iF ];,,u DDDDDDDDDDDDDDDDDDDDDD8kdk3iD

XI6[!?
{ 9kdI6[-hk’muwj fhc8h,16jI6[ X!? 9qj’c8h,I6[-hk’faj’djk;.lIj 6[lujc9lkdmuj,u 10×6
8kdk3i X8kdk3iI6[ X@? c]t I6[ X!? ,u0t|kfgmqjkdao?D
{ 9eo;o8kdk3i I6[X@?,u9eo;oDDDDDDDD8kdk3i c]t ,uDDDDDDDDgmjnvohvpdj;kI6[X!?D
{ 9qj’;afcmd];’pk;0v’mvj o-jn8jk’M muxj tdv[gxao38-hk’.oI6[ X!? c]t I6[

X@? c]h;gvqk,kxP[mP[daoD oadIPola’gdfgsaoco;.fcfj L
{ 9qj’lq,mP[,5,8kj ’M muxj tdv[gxao-hk’I6[ X!? c]t I6[X@?F oadIPola’gdf
gsaogxaoco;.f L

XI6[@?

167 

{ I6[-hk’I6[X@? ohvpd;j k I6[X!?DDDDDDDDDDDDDDDgmnjvD\kp7;k,;kj I6[-hk’I6[X@? wfhsp=h
l;j o,k9kdI6[-hk’I6[X!?D

0D .shoadIPola’gdfg[y’j I6[muj4ndc8h,.oI6[lucj 9lkdmu,j u 10×10 8kdk3i0v’
I6[ X#{$?D

{ 8kdk3i I6[X#? da[I6[X$? ,u9eo;oDDDDDDDDDDDDDDDD8kdk3igmqjkdaoD
{ .shoadIPo.-hw,h[aomafcmd0hk’0v’8kdk3iI6[X#?F cmdwfhDDDDDDDDDD mm c]t cmd

0hk’0v’8kdk3iI6[X$? cmdwfh DDDDDDDDDDDDDDDDDD mm
{ oadIPoxP[mP[ I6[X#? c]t I6[X$?DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

10

9

8 ສະຫງສ.ວວນ.ລິສຂະສິດ

7

6 10
5 9
4 8
7
3 6
2 5
4
1
1 2 3 4 5 6 7 8 9 10 3
2
I6[X#?
1
1 2 3 4 5 6 7 8 9 10

I6[X$?

7D 9kdI6[-hk’I6[X!? c]t I6[X@? ,u0=hla’gdf7n 9eo;o8kdk3i0v’I6[X@? |hvp

d;j k8kdk3i0v’I6[X!?DDDDDDDDDDgmnjv c8j0t|kf0v’8kdk3ima’lv’I6[gmqjkdaoD

c]t I6[X@? ohvpdj;kI6[X!?DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

{ 9kdI6[X#? c]t I6[X$? ,u0=hla’gdf7n 9eo;o8kdk3ima’lv’I6[gmqjkdaoc8j0t

|kf0v’8kdk3iI6[X$? cmdwfhohvpd;j k0t|kf0v’8kdk3iI6[X#? c]t

I6[X$? ohvpd;j kI6[X#?DDDDDDDDgmnjv gIqkg;qhkwfh;kj I6[X$? s=hplj;o,k9kd I6[X#?D

’D .omk’dq’dao0hk,gIqkg;qhkwfh;jk-hk’I6[X!? wfh0tspkp,k9kdI6[X@? c]t

I6[X#? wfh0tspkp,k9kdI6[X$?D

168 

dyf9tdemu!
I6[8+wxoUc,joI6[fyol;o8vo|bj’g-y’j gxaoI6[lujc9lkd,u];’pk;8q; 9y’500 m
c]t ];’dh;k’8q;9y’ 300 m D
dD 9qj’.-hw,h[aomafcmdI6[l;o8vooUG

XI6[&? ສະຫງສ.ວວນ.ລິສຂະສິດ

{ ];’pk; AB .oI6[X&? cmdwfhDDDDDDDDDDD cm F mP[gmqkj da[];’pk;8q;9y’DDDDDDDDDDD cm
{ ];’d;hk’ BC .oI6[X&? cmdwfh DDDDDDD cm F mP[gmqjkda[];’d;hk’8q;9y’DDDDDDD cm
0D 4k,;jk];’pk; AB F ];’d;hk’ BC 0v’I6[fyol;oohvpd;j k8q;9y’9adgmnjv L
{ .shoadIPola’gdfg[yj’8k8t]k’]5j,oU c]h;8jn,.lj[jvo9Eg,af.sh4nd8hv’D

fyol;o I6[c8h, 8q;9y’ ];’pk;.oI6[c8h,sp=h ,kf8klj;o
(cm) (cm) 9kd8q;9y’DDDDDDDDgmnjv

];’pk; AB ........ ........ .............. 1
0hk’dh;k’ BC ........ ........ .............. .............

1
.............

169 

7D 0h=la’gdf
{ 9kddko7yfw]j.o4aol5fmhkp8k8t]k’0v’0=hX0? gsao;kj ];’pk; AB F BC

0v’I6[c8h,ohvpdj;k];’pk;8q;9y’ 5000 gmnjvD
{ dkoc8h,I6[.sh0t|kf|hvpd;j k8q;9y’gvuho;kj dkos=hplj;o 9kd8q;9y’8k,

,kf8kl;j ofaj’oUG
];’pk; 1cm 0v’I6[c8h,mP[gmqkj 5000cm 0v’];’pk;8q;9y’0Pogxao

,kf8kl;j o 1 s^n 1:5000 D

5000

@D .97;k,

 ,kf8kl;j o0v’czoza’gmqjkda[zqosko0v’];’pk;0v’I6[c8h,

.sh];’pk;8q;9y’D

 0Pogxaog]dl;j oG ,kf8kl;j o = ];'pk; AB0v'I[6 c8,h
];'pk; AB0v'8;q 9y'
ສະຫງສ.ວວນ.ລິສຂະສິດ
X7yfw]sj q;|j;pvaofP;dao?

8q;1jk’ !G

9qj’8nj,.l[j vj o9Eg,af ];’pk;8q;9y’
(cm) 700 1200
{ ,kf8kl;j os=hpc,jo 1 × 1  
100 ];’pk;0v’I6[ 100
g-yj’gxaog]dl;j omuj,u9eo;o c8h, ( cm)
....... 12
r6fohvpdj;kr6f ltoaho7kj

0v’,kf8kl;j o8hv’ohvp

dj;k1D ];’pk;8q;9y’

8q;1jk’ @G (mm) ....... 70

9q’j 8jn,.lj[jvo9Eg,af ];’pk;0v’I6[ 200 350 ×5  
c8h, ( mm )
{ ,kf8klj;o0tspkp

c,jo 5 g-yj’gxaog]dl;j omuj
1

9eo;or6fs^kpd;j kr6fD ltoaho7kj 0v’,kf8kl;j o8hv’s^kpd;j k1D

170 

-vdsk,kf8kl;j omujg\ktlq,

vu’8k,xN,0Po 0v’oadIPoc8j]t7qoF .shoadIPo .-hw,h[aomafcmd0t\kfxN,

0v’c8j]t7qo9trq[;jkG

{ ];’pk;0v’\hkg9hpcmdwfh DDDDDDDDDDDDDDDDDDDDDDDDDDDDDD cm

{ ];’dh;k’0v’|hkg9hpcmdwfh DDDDDDDDDDDDDDDDDDDDDDDDDDDD cm

 8k,|hkg9hp0v’oadIPomuj,u16j r;doadIPolk,kfc8h,I6[fyol;omuj,u];’

pk; 8q;9y’ AB=500 m c]t ];’dh;k’8q;9y’ BC=300m D 8k,,kf8kl;j o

1 wfh[+ L phvospa’ LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
1000

4hkg]nvdgvqk,kf8kl;j o 1 r;doadIPolk,kfc8h,I6[.\j.l|j hkg9hpmu,j u16jwfh
4000
s^n[+ LDDDDDDDDDDDDDDDDDDDDDDDDphvospa’ LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

#D [qfg/ydsaf

!D w]ptsjk’8q;9y’c89j 5f A sk95f B cmdwfh 100 km .sh-vdsk,kf8kl;j o0v’
c8]j tczomuj muj8kj ’daofaj’oUG
{ .oczomujmu! gryjoc8h,95f A sk95f B skj ’dao 1cm
{ .oczomujmu@ gryoj c8h,95f A sk95f B sjk’dao 50 mm
{ .oczomumj u# gryoj c8h,95f A sk95f B sjk’dao 0,125m
ສະຫງສ.ວວນ.ລິສຂະສິດ

@D 9q’j la’gdfg[y’j 8k8t]k’]5,j oUFc]h;8jn,9eo;omuj4nd8hv’.lj

w]ptsjk’16jczomuj w]ptsjk’8q;9y’ ,kf8kl;j o
1:100 000 s^n 1
2 cm 2 km
100 000
10 mm 10 km
1cm 4 000 m DDDDDDDDDDDDDDDDDDDDDDDDDDD
2, 5 dm 125 hm DDDDDDDDDDDDDDDDDDDDDDDDDDD
5m 25 km DDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDD

171 

#D dtfkofe czoj |b’j gxaoI6[lujc9lkdmuj,u];’pk; 4 m c]t ];’dh;k’ 2 m
4hkoad IPo8hv’dkoc8h,dtfkofeczjooAo.lxj N,D

dD 4k,;kj oadIPo8hv’c8h,dtfkofeczoj oAogxaoI6[-toyf.f L
0D 4hkoadIPoc8h,.ljxN,F 8k,];’pk;0v’dtfko10cm D 4k,;kj oadIPowfh.-h

,kf8klj;ogmqkj .fL
7D 8k,,kf8kl;j omujwfh.o 0=hX0? .shoadIPo[vd;kj 9tc8h,];’dh;k’0v’dt

fkooAo9ad cm L

$D .oczomuj[k’xtgrf gryjo,ad\kp,kf8klj;o3fp.-hI6[0ufgxaomjvo-jn c]h;

\kpw]ptsjk’8q;9y’.l0j uf\kp0v’mvj o-jnfaj’]5j,oUG

I6[mjvo-jn 7evtmy[kp ,kf8kl;j o

0 5km 10km 1cm mP[gmqjkw]pt 1
sjk’8q;9y’ 5km 500 000

DDDDDDDDDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
ສະຫງສ.ວວນ.ລິສຂະສິດ
0 50 100 150m

DDDDDDDDDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

0 25 50km

DDDDDDDDDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

0 45 90km

9qj’.-hw,h[aomafcmdw]ptskj ’0uf\kp0v’c8]j tmvj o-njF c]h;8jn,.lj[jvos;kj ’
X.o8k8t]k’0hk’gmy’? .sh4nd8hv’D

&D 9qj’-vdsk,kf8kl;j o.oc8jd=]tou]5j,oUG
dD w]ptskj ’.oczomu0j v’ 95f A sk 95f B c,oj 3cm c8wj ]ptsjk’8q;9y’

c,jo 12km D
0D .o|hkg9hpczoj |b’j gryjoc8h,I6[mu,j u];’pk;c89j 5f A 95f B c,oj 18cm c8]j ;’

pk;8q;9y’c,oj 2cm

*D fyol;o8vo|bj’ gxaoI6[lujc9lkdF ];’pk;8q;9y’cmdwfh 200m c]t ];’
dh;k’ 140m .shc8h,fyol;o8vooAo.ljxN, -b’j ];’pk;0v’fyol;oc,oj 5cm D
X0=hcotoeG 8hv’-vdsk,kf8kl;j oc]t-vdsk];’dh;k’0v’fyol;o.oI6[?

172 

(D I6[lucj 9lkdoUwfh4ndc8h,3fp.-h 40m
,kf8kl;j o.f[+ L 4hkc,jo.sh
-vdsk,kf8klj;ofa’j dkj ;D 15m

)D .ow]ptsjk’8q;9y’0v’g,nv’ A c]t g,nv’ B c,oj 420km wfh4ndc8h,16j.oczo
muj mu,j uw]ptskj ’84mm .sh-vdsk,kf8kl;j oD

_D .oczoza’|bj’muj,u,kf8kl;j o 5cm .o|hkg9hpmP[gmqkj da[ 2km 8q;9y’
dD 9qj’-vdsk,kf8klj;o0v’czoza’D
0D .sh-vdsk];’pk;0v’gfuoj [yo.oczoza’.og,njvgfuoj [yooAopk;8q;9y’ 8 km D

!WD .shc8h,I6[lk,c9Flk,0hk’cmdwfh 3cm;4cm c]t 5cm D
dD g[y’j 8k,I6[c8h, I6[fa’j djk;c,joI6[lk,c9-toyf.f L
0D gryjo0tspkpI6[lk,c9faj’dkj ;fh;p9advafgvdtlko 0hk’mujcmdwfh 5cm dkp

gxao 10cm D 4k,;kj lv’0hk’mujpa’gs^nv9tcmdwfh9ad cm L
7D c8h,I6[lk,c9.\mj u0j tspkpc]h;D
’D I6[lk,c9.\oj Uc,oj I6[lk,c9-toyf.fL c]t wfh0tspkp9kdI6[gdqjk9adgmjnvL

!!D I6[lujc9lkdmucj 8h,16j0hk’]5,j oUwfh4ndc8h,3fp.-h,kf8kl;j o[+ L phvospa’ L

40 m 
ສະຫງສ.ວວນ.ລິສຂະສິດ

20 m

!@D g9hpc0’czjo|bj’gxaoI6[lucj 9lkd0t|kf 2 m c]t 0,8m D gryjo8hv’dko

8afg9hpc0’czoj oAo gxaoI6[9t85]afmuj,u0hk’cmdwfh 20cm D

dD 4k,;jk gryjo9twfhg9hpmugj xaoI6[9t85]af9adczjo L

0D .shc8h,I6[ 3fp.-h,kf8kl;j o 1 D
20

173 

!#D grnvj vhv,Iqh;fyo8vo|bj’mujgxaoI6[9t85]afF gryoj wfh9kj pg’yo7jk;af45 c]t cI’
’ko9eo;o1641000 du[D Ih6;jk 7kj vhv,Iqh; 1m 8qdgxaog’yo11400 du[D
4k,;kj 0hk’0v’fyo8vofa’j dkj ;,u9ad m L

!$D gryjogIafIqh;vhv,|v’|;j p|bj’muj,u];’pk; 15,5m c]t 10,5m ];’dh;k’

fh;ps^adglqkw,hskj ’dao 2,50 m c]t gryoj wfh/a’glqkw,hsjk’9kdc7,|v’c8j]t

g[Nv’ 3,50 m c]t c8j]tc90v’Iqh; gryoj wfh/a’glqkw,hc9]t 1 s^adD

dD .shc8h,I6[3fp.-h,kf8klj;o 1
100
0D 4k,;kj 9eo;os^adglqkw,h,ugmqjk.f L

!&D gryjo0v[mk’1k’fh;pmvj o-u,a’lts^adI6[ pk; 1,20 m D mk’1k’8vooAo

cmdwfh 432 mm .oI6[c8h,muj,u,kf8klj;o 1 D

20 000

dD 4k,;kj ];’pk;0v’mk’1k’8vooAo,u9ad m L
0D 9eo;omjvo-u,a’muj9tgvqk,k0v[mk’,u9admvj oF 4hkgryjo0v[mk’1k’c8jrP’

g[Nv’fP; L
7D mjvo-u,a’mjvo|bj’,uoE|ad 60 kgF gryoj 0qomjvo-u,a’ma’\qf,kpa’[vj o0v[mk’

1k’F gryoj 8hv’.-h]qf0qolqj’mujlk,kf0qowfh4hP;]t 3,6 38oD 4k,;kj gryjo9t0qo
wfh9ad4hP; L
ສະຫງສ.ວວນ.ລິສຂະສິດ

174 

[qfmu @@ ,kf8klj;o X8+?

!D dyf9tde

dyf9tde ! Xm;o7no?

dD .sh8nj,.l8j k8t]k’ 1,2,3 ]j5,oU

8k8t]k’ ! 8k8t]k’ @ 8k8t]k’ #

×2 ÷5 ÷5 ×2 ×0,4

1 ..... ..... 1 ..... ..... 1 ..... 
5 ..... 
5 ..... ..... 5 ..... ..... 6, 4 ..... 

6, 4 ..... ..... 6, 4 ..... ..... 9 ..... 

9 ..... ..... 9 ..... .....

ສະຫງສ.ວວນ.ລິສຂະສິດ
 9qj’lq,mP[zqowfhIa[4aol5fmhkp.o8k8t]k’ 1, 2 c]t 3D
{ .o8k8t]k’!F gryoj 76o9eo;o0v’4aomu|b’j da[ 2 c]h;sko9eo;omujwfh,k
0v’4aomulv’.sh 5D
{ .o8k8t]k’ @F gryoj sko9eo;o0v’4aomu|bj’.sh5 c]h;76o9eo;omuwj fh,k
0v’4aomulv’da[ 2

{ .o8k8t]k’#F gryjo76o9eo;o0v’4aomu|b’j da[ 0,4 Xs^n 2?D

5

0D 8q;1kj ’ .sh7yfw]j 25×21 ; 63×8 ; 21× 4

7 7 17

;ymucdhG 25× 21 = 25×( 21÷ 7 ) = 25×(......) =......................

7

63× 8 = ( 63÷ 7 )×8 = (............)×8 =......................
7

22,1× 4 = ( 22,1× 4 ) ÷17 =.............÷17 =......................
17

175 

0=h7;o9nj

le]a[dko76o9eo;o d  .sh a (b ≠ 0) F gryoj lk,kf7yfw]jfaj’oUG

b

{ 76o d  da[zqosko a:b
{ 76o d da[ a c]h;skozqoIa[.shb
{ d sko.sh b c]h;76ozqoIa[da[ a

d× a = d×(a÷b) = (d×a)÷b = (d ÷b)×a
b

dyf9tde @

dD .sh7yfw]j 9,3× 2 ; 6,3× 2 ; 3×5 ; 11× 3 ; 1 ×9
3 33 11 9

0D 3 0v’ 750 du[ c,oj 4 ×750 = ……………………………………………………………………..

45

7 0v’ 60km DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

5
ສະຫງສ.ວວນ.ລິສຂະສິດ
3 0v’ 120D ……………………………………………………………………………………………………

4

dyf9tde #

.shoadIPola’gdfg[y’j czoza’0v’gInvo]5j,oU c8]j t0hk’ a,b,............., h  wfh\kp
];’pk;8q;9y’gxao cm D 8q;1jk’G a = 660cm .shoadIPo.-hw,h[aomafcmd];’pk;
0v’ a,b,c,.................., h .oI6[czoza’ 9kdoAo8jn,.lj8k8t]k’]5,j oU.sh7q[D
 I6[czoza’ d=144 c=516

f=192

h=528

e=336

b=372 g=180

a=660

176 

 8k8t]k’

];’pk;9y’ (cm) a b cd e f gh
÷.... ];’pk;.oI6[ (cm)
660 372 ... ... ... ... ... ... ×....

5,5 ...... ... ... ... ... ... ...

{ 9q’j 7yfw]zj qosko];’pk;0v’I6[.o.sh];’pk;8q;9y’.oc8]j t4aoD oadIPola’
gdfzqoskomuwj fh,kgxaoco;.f LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

×.... ÷....
{ 9q’j xtdv[ 5,5 = 55 = 1 faj’oAo 5,5 = 1
660 ..... ..... 660 .......
×.... ÷....
 

{ vaf8klj;o 1 gvUo;jk co;.fle]a[czoza’ L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

120

{ ];’pk;8q;9y’76o.sh,kf8kl;j o c,jo];’pk; DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
ສະຫງສ.ວວນ.ລິສຂະສິດ
@D .97;k,

];’pk;.oczoza’= ];'pk;8;q 9y'× ,kf8kl;j o 
Xsq;|;j p;afcmdvaofP;dao?

8q;1jk’ !D
w]ptsjk’c8j8ts^kfg-qhkskgfujody]kcs’j -kf,u 15km  4hkoadIPo c8h,muj8A’0v’

lv’95f.ljg9hp8k,,kf8kl;j o 1 4k,;jkoadIPo9tc8h,lv’95foAoskj ’dao

100 000

gmqjk.fcm L
cotoedko7yfw]j
.shxPj ow]pt8q;9y’ 15km = .........................cm
vu’8k,l6f7yfw]Gj
w]ptsjk’.oI6[c8h, = w]ptskj ’8q;9y’ × ,kf8klj;o

177 

w]ptskj ’.oI6[c8h, = ........................× 1

100 000

= ....................×1
100 000

= (....................) ÷100 000 = .........................

ltoAo w]ptskj ’.oI6[c8h,c,jo DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD cm L

8q;1jk’ @D
dD .shoadIPola’gdfg[y’j I6[ X&? .o[qfmuDDDDDDDDDD|hkDDDDDDDDDD I6[fyol;o8vooAosp=hlj;o

,k9kdfyol;o8q;9y’5000gmjnv s^ng;qhk;kj dkoc8h,fyol;oI6[ X&? r;dgIqkwfh

.-h,kf8klj;oG 1

...................

{ .sh.-hw,h[aomafmu,j u16jcmdc89j 5f I sk95f j Xw]ptsjk’8Qow,hlv’8Qo?,u DDDDDDDDDDDDcm
{ 4k,;jk w]ptskj ’8q;9y’0v’lv’8Qow,h,u9ad m L X.sh-vdsk.o8k8t]k’]5j,oU?D
 8k8t]k’

•  9qj’8nj,.lj[jvo9Eg,af.sh4nd8hv’D

×........
ສະຫງສ.ວວນ.ລິສຂະສິດ
fyol;oI6[X&? I6[c8h, ];’pk;9y’ ,kf8klj;o
X cm ? X cm ?
];’pk; 1
10 ............ 5 000
];’d;hk’
............ 30 000 1
w]ptskj ’]ts;jk’ ............ ...........
95f I sk95f J
............ 1
...........

×........

•  w]ptsjk’]ts;jk’95f I sk95f J  .oI6[c8h, 5,5cm 76o.sh9eo;oDDDDDDDDDDD c,jo
w]ptskj ’8q;9y’0v’ IJ  gmqkj da[DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

•  w]ptsjk’ IJ  .oI6[c8h,5,5cm sko.shDDDDDDDDDDDc,jow]ptsjk’8q;9y’0v’ IJ DDDDDDD

178 

;ymucdhG

5,5 ÷ 1 = a ⇒ 1 × a = 5,5 ⇒ a = 5,5
5 000 5 000 5 000

faj’oAo a×5000 ÷5000 = 5,5×5000 ⇒ a = 5,5×5000 = 27 500

a = 5,5 ÷ 1 = 5,5×5000 = 27 500
5 000

lts^5[

w]ptskj ’8q;9y’ = w]ptsjk’.oI6[c8h, ÷ ,kf8klj;o
s^n w]ptskj ’8q;9y’ = w]ptskj ’.oI6[c8h, × 8q;xUo0v’,kf8klj;o 

8q;1jk’ #D ສະຫງສ.ວວນ.ລິສຂະສິດ
.oczomujxqd7v’0v’ lD xD xD ]k;F gryoj .sh,kf8kl;j o 1:1500000 , w]ptsjk’ 
c8jc0;’s^;’rt[k’skot7vos^;’cmdwfh 14,5cm  4k,;jkw]ptskj ’8q;9y’c8j
s^;’rt[k’skot7vos^;’9t,u9ad km  L

cotoedko7yfw]j
dD w]ptskj ’8q;9y’ = w]ptskj ’.oI6[c8h, × 8q;xUo0v’,kf8klj;o

= 14,5cm× ..............
1

= 14,5cm×..............
1

= 14,5cm×..............  

= ........................cm

= ........................km

#D [qfg/ydsaf

!D 16j.oczoza’muj,u,kf8kl;j o 1 , w]ptskj ’c8gj ,nv’ A skg,nv’ B  cmd

5000 000

wfh 134mm  gInv[yo]e|bj’vvd9kdg,nv’ A  8vo!@ 3,’ 20 okmuD 4k,;jkgInv
[yo]eoAo9tIvfg,nv’ B  9ad3,’F 4hkgInv[yo]efa’j djk;,u7;k,w;ltg]pj
134km  8+ ! -q;3,’D

179 

ສະຫງສ.ວວນ.ລິສຂະສິດ@D 16j95f37h’0v’mk’glAo|b’j gryoj wfhxads^adglqkohvpgrjnvlaopkog8nvozh60a[0u.j sh]t
;a’D w]ptsjk’0v’lv’s^adglqkc,jo 1,20m D 1j6.oczoza’muj,u,kf8kl;j o
1 , w]pt37h’0v’mk’1j695ffa’j djk;cmdwfh 43,20mm .

20 000

dD .sh-vdskw]ptskj ’8q;9y’0v’95f37h’oAo
0D 9eo;os^adglqkohvpmuj9t.-h,u9ads^ad L
7D c8j]tglqkohvp,uoE|ad 60kg. 4k,;jk 9t.-h]qf9ad4hP; grjnvgvqks^adglqkohvp

,k[jvo95f37h’F g,njvIh6;kj c8]j t4hP;0v’]qf0qolq’j lk,kf0qowfh 3,6 38oD

#D c];lkpw2cI’l6’csj’|b’j xtdv[fh;plkpw22hklk,lkpmugj oaj’9kd95f A  
sk95f B . lkpw22hkfa’j dkj ;wfh1v’1j6gmy’s^adglqkF w]ptskj ’0v’lv’s^adglqk
c,jo50m . gryoj wfh8A’s^adglqks^ad|b’j 16j95fA c]tvuds^ad|b’j 16j95fB . w]ptsjk’16j 
.oczoza’c8j95f A  sk95f B . cmdwfh 239mm , 8k,,kf8kl;j o 1

50 000

dD .sh-vdskw]ptsjk’8q;9y’c89j 5f A sk 95f  B  cmdwfh9ad m  L
0D .sh-vdsk];’pk;0v’lkpw22hkma’\qf L
7D .sh-vdsk9eo;os^adglqkma’\qf L

$D gryjoc8h,I6[.oczoza’|b’j 3fp7yfw]jgvqk];’pk; 1cm  mP[gmqkda[];’pk;
      1000m .  4k,;jk ,kf8klj;o0v’czoza’oAo,ugmqjk.f L

&D gryoj 8hv’dkoc8h,I6[lk,c9mucj 8]j t0hk’cmdwfh3m, 4m,5m , 8k,]efa[D 4k,
;jk .oczoza’mu,j u,kf8kl;j o 1 , c8]j t0hk’0v’I6[lk,c9,u9ad cm L

100

*D czomujmk’cs’j |b’j muj,u,kf8kl;j o 1 .

200 000

dD 4k,;jk mk’.oczomucj mdwfh 1cm F 5cm  8q;9y’9tcmdwfh9ad m  L
0D w]ptskj ’8q;9y’0v’lv’g,nv’,u 15km D 4k,;jk w]ptsjk’0v’lv’8q;g,nv’

.oczomuj9t,u9ad cm L
(D 16jg[Nv’]5,j 0v’czomurj 6,lkf0v’xtgmf gryoj gIafg7njv’\kpfa’j ]5j,oUG

150 km

9q’j -vdsk,kf8klj;o0v’czomuoj UD

180 

)D .shczoza’|b’j muj,u,kf8kl;j o 1 .

50 000

dD 9qj’-vdskw]ptsjk’.oczoza’gxao cm  mujmP[gmqjkda[w]ptskj ’8q;9y’ 500m F
       4 km; 3,2 km  8k,]efa[D

0D 9q’j -vdskw]ptsjk’8q;9y’gxao km ,mujmP[gmqjkda[w]ptskj ’.oczoza’c,jo
30mm,   6cm, 11cm 8k,]efa[D

_D .o|hkg9hp0v’I6[|b’j F gryoj cmdwfh 7,5cm mP[gmqjkda[];’pk;8q;9y’ 3mm .
dD 9qj’-vdsk];’pk;.o|hkg9hp mujmP[gmqkj da[];’pk;8q;9y’6mm, 8mm ,

8k,]efa[D

0D 9qj’-vdsk];’pk;8q;9y’gxao mm, le]a[];’pk;16j.o|hkg9hp5cm, 2,5cm
\kpgsfG [qfg/ydsaf[qfoUc,jog;qhkg4y’dko0tspkpl;j oD

!WD ,kf8klj;o0v’czoza’|bj’c,oj 1  grnjv.shcmfg\ktda[8q;9y’ gryoj 7yf1kd

24

.fhczoza’.\jF 3fpdko8nj,g0qhk.lj 1 0v’czoza’gdqkj D 4k,;kj ,kf8klj;o

3

0v’ czo za’.\cj ,jogmqkj .f L
ສະຫງສ.ວວນ.ລິສຂະສິດ
!!D ,kf8klj;o0v’czoza’|bj’c,jo 1 gryjo1kdlhk’czoza’.\Fj 3fpdkogvqk
60

vvdc8]j t0hk’ 1 0v’czoza’gdqkj D 4k,;jk ,kf8klj;o0v’czoza’.\j
4
c,jogmqjk.f L
!@D 60 mm

dD 9qj’c8h,I6[lujc97no.\3j fp 25 mm
c8h,8k,0t|kfmuj\kpw;h

.oI6[D .shoadIPo.-hw,h[ao

mafcmdsk];’pk;0v’glAogoaj’9v,D

0D I6[lujc9lkdoUwfhc8h,9kdmqj’ok8q;9y’F 8k,,kf8kl;j o 1 D 9qj’-vdsk

10 000

0hk’pk;F 0hk’d;hk’ c]t ];’pk;0v’glAogoa’j 9v,8q;9y’0v’mq’j okoAoD

!#D .oczomu|j bj’F gryjo.-h,kf8klj;o 1 D

400 000

dD 9qj’-vdskw]ptsjk’.oI6[c8h,0v’lv’g,nv’muj,uw]ptsjk’8q;9y’ 24km .
0D 9qj’-vdsk];’pk;8q;9y’0v’glAomk’muj4ndc8h,.oI6[c8h,16cm

181 

!$D ]qfgda’7ao|b’j ,u];’pk;5,21m F ];’d;hk’1,87 m  c]t ];’l6’1,22m D gryoj

8hv’dkoc8h,I6[8k,,kf8kl;j o 1 D 9qj’-vdsk0t|kf0v’I6[]qfgda’faj’

100

dkj ;.oI6[c8h,D
!&D dD 9q’j 8jn,.lj[jvo9Eg,af.sh4nd8hv’0v’xt3spd8+wxoUG

<,kf8kl;j o 1 <

200 000

1cm mucj mdwfh DDDDDDDDDDDDDDDDDDDDDDDD mP[gmqkj da[DDDDDDDDDDDDDDDDDDDDDDDDDDDD s^n DDDDDDDDDDDDDDD km
0D .o8k8t]k’]5,j oU .sh8nj,.l0j =h,6o.sh4nd8hv’F 3fp7yfw]j8k,,kf8kl;j omuj[vd.o

0=h dD gmy’oUD
w]ptskj ’.oI6[c8h, w]ptsjk’8q;9y’ w]ptskj ’8q;9y’

X cm ? X cm ? X km ?

4 ................. .................

16 ................. .................
ສະຫງສ.ວວນ.ລິສຂະສິດ
................. 2 000 000 .................

................. ................. .................

!*D ]qf7ao|b’j mucj ]oj vvd9kdg,nv’ A  skg,nv’ B  fh;p7;k,w;ltg]jp 44 h D

.oczomju mu.j -h,kf8klj;o 1 w]ptsjk’c8j A  sk B  cmdwfh 176cm D

200 000

dD 9qj’-vdskw]ptsjk’8q;9y’c8gj ,nv’ A  skg,nv’ B D
0D 9q’j 7yfw]jskg;]kmu8j hv’w-hc]jo c8gj ,nv’ A  skg,nv’ B  0v’]qf7aofa’j dkj ;D

182 

ສະຫງສ.ວວນ.ລິສຂະສິດ[qfmu @# dj5, c]t vq’xtdv[0v’dj5,

!D leo;o]5,j oU,u7;k,\kpco;.f L

dv’suoF \;f8=ks^;fdv’7tfuF dkoltlq,ltc8,F /6’oqdF lt,k7q,oad8bdgxafF
7tot]t7volaf DDD
 leo;o0hk’gmy’oU\kpg4y’G dko9aflyj’0v’F ;af45F 7qoF laf DDDD 1j6Ij;,daogxao
\;fF \6jF95F r;dF 7totF /6’ DDD -bj’vkf9t,u7;k,leraodao s^n [+,u7;k,lerao
daoD oad7toyflkfwfhg]nvd.-h7emu,j u7;k,\kpdk’7nG d5,j mu1j 6jI;j ,daoc8]j tvao.o
d5j,oAogvUo;kj vq’xtdv[ 0v’d5j,D

dyf9tdemu !

dD .shoadIPo8v[7e4k,]5,j oUG
 lt,k7q, “vk-Po” ,ult,k-yd9adxtgmf L 7nxtgmf.fcfj LDDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
 xtgmf]k; c,oj gxaolt,k-yd0v’lt,k7q,vk-Po[+ LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
 gIqkg;hqk;jk xtgmf]k;c,oj vq’xtdv[ 0v’ d5j,vk-PoD 4hk\kpxtgmf]k;
fh;p 8q;vadlvo A c]t d5j,vk-Po\kpfh;p8q;vadlvo A F 0Po;jk A c,oj vq’
xtdv[ 0v’dj5, A D
 9uo \kpfh;p8q; c c,oj vq’xtdv[0v’d5j, A [+ LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
gIqkg;hqk;kj G c [+c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD0v’d5,j A D
0D S c,jod5j, g9af;ao.o|b’j vkmyf. 9qj’0Po.l[j jvo9Eg,af]5j,oU.sh4nd8hv’D

;aor5f (w) c,oj vq’xtdv[0v’ S ; 0Pogxao w∈S
;aoxn.\j (n) [+c,jovq’xtdv[0v’S ; 0Pogxao n∉S
;aovkmyf(d ) DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD S ; 0Pogxao d .......S
;aoglqk (a) DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD S ; 0Pogxao DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
;ao9ao (m) DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD S ; 0Pogxao DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
;aortsaf (t) DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD S ; 0Pogxao DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
gfnvo,a’dvo( j) DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD S ; 0Pogxao DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

183

{ .sh-vdskvq’xtdv[0v’ S mu[j +wfhdjk;.o0hk’gmy’oUG c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

{ oadIPolk,kf3I,gvqkvq’xtdv[0v’d5j, S 3fp0Pofh;pI6[c[[oUG

S = { rf5 F vkmyfF 9oa F ld5 F rtsfa F glkq F v'a 7ko}

7D .shoadIPola’gdfg[y’j I6[8+wxoU 9ao DDDDDDDDDDDDDDDDDDDDDDDDDDD S
c]h;0Povq’xtdv[0v’dj5,muj l5d DDDDDDDDDDDDDDDDDDDDDDDDDDD
pa’[+mao7q[D DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

dyf9tdemu @ D
.shoadIPola’gdfg[y’j I6[dj5,lk,I6[8+wxoUD

MC

ສະຫງສ.ວວນ.ລິສຂະສິດ
dko0Po8+wxoU c,oj dko0Po[k’d5j,zyfG

d5j, M 0Pogxao {a , , , a ,U, 2, 2}

d5j, C 0Pogxao {3,3,3}
d5,j D 0Pogxao { a , d ,b , c , e ,t }

faj’oAo 9q’j d;f7no c]h;0Po.sh4ndG

M = { ................................... }

C = { ...................................}

D = {a,d ,b,c,e,t }

dyf9tdemu #

dD { 9qj’la’gdfg[yj’d5j, A = { 0F lF 4F zF /F s} 8q;vadlvo 0F lF 4F zF /F s

]h;oc]h;c8jgxaovadlvoDDDDDDDDDDDDDDDDD.ortpao-tot]k;D d5j, A c,jo “dj5,vad
lvoDDDDDDDDDDDDDD.ortpao-tot]k;”

{ dj5, B = { dF 9F fF 8F [F xF 1F v} B c,jodj5, DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

184

ສະຫງສ.ວວນ.ລິສຂະສິດ{ C c,jod5j,8q;g]d.-h0Po9eo;ome,t-kf 275 9q’j 0Pod5j, C = { .......}
{ D d5j,8q;g]d.-h0Po9eo;ome,t-kf 42124 9qj’0Pod5j, D = { .......}

@D .97;k,

dko0Pod5j,8hv’,uG
{ ;q’xuddk
{ 95fs^n9E95fgrnjv0Aovq’xtdv[lv’8q;
{ vq’xtdv[|b’j 0PowfhgmnjvfP;.od5j,
{ vaofa[0v’dko0Povq’xtdv[[+le7ao

9kddkoIPodjP;da[dj5,zkj o,kFgIqkdeoqfdj5,wfhlv’c[[7nG
!D c[[ c9d1kp vq’xtdv[D

8q;1kj ’G A = { 0F lF 4F sF /F z} gvUo;kj dkodeoqfd5j,c[[ 0tspkp vq’xtdv[

@D c[[g;qhkg4y’75o]adltot];, 0v’vq’xtdv[0v’d5j,D
8q;1jk’G A c,jodj5, “vadlvol6’” gvUo;kj dkodeoqfd5j,c[[ sp=h

#D [qfg/ydsaf

!D d5j, E c,jod5j,0v’gfnvomu,j u 31;aoD 9qj’0Pod5j, E c[[0tspkp

E = { ,a’dvoF ,uokFDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD }

@D { ` c,jodj5,9eo;ome,t-kfD 9q’j 0Pod5,j ` c[[0tspkpD

` = { 0,1, 2,................................... }

{ gIqklk,kfc9d1kpvq’xtdv[0v’d5j, ` wfh\qfm5dM8q;[+ L phvospa’ L
{ gIqkgvUodj5,9eo;ome,t-kf ` gxao dj5,[+lyhol5f
{ 0 c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD `F 8,54 [+c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 0v’ `D

185

#D la’gdfg[yj’I6[d5j,8+wxoUD S V
P

{ d5j, P ,u9advq’xtdv[ L P = { .................} P gvUo;jk d5j,76jສະຫງສ.ວວນ.ລິສຂະສິດ

{ dj5, S ,u9ad vq’xtdv[ L S = { .................} S gvUo;jk d5j,gvd

{ dj5, V ,u9advq’xtdv[ L V = { } V gvUo;jk d5j,gxqjk laopt]adfh;p V = φ

$D la’gdfg[yj’8q;1kj ’]5,j oUg-y’j c,oj dj5,fP;daoc8j0Povvdgxaolv’c[[7nG
8q;1kj ’ !G

S = { 9aoF 7koF r5fF rtsafFld5 F glkq F myf} 4k,;kj d5,j S deoqf.oc[[.f L

S c,oj dj5, “ g9af;ao.o|bj’vkmyf ” 4k,;jk d5j, S deoqf.oc[[.f L
8q;1kj ’ @G G
M c,oj d5j,0v’gfnvomuj,u 30 ;aoS M deoqfd5j,c[[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

M = { g,lk DDDDDDDDDDDDDDDDDDDDDDDDDDDDD } S M deoqfd5j,c[[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

&D dD .sh0Pod5,j 8+wxoU c[[0tspkpvq’xtdv[D
A c,jo9eo;ome,t-kfmusj ^kpdj;k 3 c]t |hvpdj;k 13 D
P c,jo9eo;ome,t-kf76mj usj ^kpdj;k 3 c]t|hvpd;j k13
I c,jo9eo;ome,t-kf7ud1j6]ts;kj ’ 3 c]t13

0D .sh\kp4nds^nzyf.ljxt3spd]5j,oUG
{ m5dM vq’xtdv[0v’ P c,oj gxaovq’xtdv[0v’ A DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
{ m5dM vq’xtdv[0v’ I c,jogxaovq’xtdv[0v’ A DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
{ m5dM vq’xtdv[0v’ A c,oj gxaovq’xtdv[0v’ P DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

*D { 9q’j 0Po d5,j 8q;g]d.-h0Po9eo;ome,t-kf 72 D
{ .sh0Po d5j,8q;g]d.-h0Po9eo;o4h;o 727; 122 436; 2 221435660.

(D { 9q’j -vdsk9eo;ome,t-kfmu|j hvpd;j k100 g-yj’d5j,8q;g]d.-h0Poc,oj
{ 2;9 } D

{ 9q’j -vdsk9eo;ome,t-kfmu|j hvpd;j k 400 g-yj’dj5,8q;g]d.-h0Poc,oj {2;9}D

186

)D { dj5,gxjqkc,oj dj5,mu[j +,uvq’xtdv[s^n9eo;ovq’xtdv[0v’d5j,gmqjkl6oxt3spd

oU4nd s^n zyf L

{ dj5,0v’gfnvomu,j ulk,ly[lv’,NFdj5,fa’j dkj ; gvUo;jk dj5,spa’ L

_D { 9q’j 0Pod5j, A; B;C; D c[[ CA
0tspkpD

{ -vdskvq’xtdv[Ij;,0v’ m D
A c]t B D B
{ -vdskvq’xtdv[Ij;, a hd j
0v’ A c]t C be
{ -vdvq’xtdv[mu[j +gvqkvq’
xtdv[0v’ C D c

k

!WD I c,jod5,j 9eo;ome,t-kfg-yj’gxao9eo;o7ud|hvpdj;k 15 D 9q’j 0Pod5j, I
c[[0tspkpD

!!D gryoj .shd5j, T = { 0F lF 4F zF /F s }

W = { dF 9F fF 8F [F xF 1F v }
ສະຫງສ.ວວນ.ລິສຂະສິດ
P = { 0, 2, 4, 6,8,10,12 }D 9qj’0Pod5j,g]qjkoUc[[sp=hD

!@D 9qj’la’gdfg[yj’I6[dj5, A c]t 8q;vadlvo a,b,c,d,e c]h;[vd;kj 8q;vadlvo

c A
a
b

e

d

.fgxaovq’xtdv[0v’dj5, A L ,u8q;vadlvo8q;.f[+c,oj vq’xtdv[0v’dj5, A L
!#D .sh-vdsk9eo;ome,t-kf mu1j 6j]ts;jk’100 c]t 400 g-y’j d5j,8q;g]d.-h

0Poc,oj { 7;3 }D

!$D { .sh0Pod5j,8q;g]d.-h0Po9eo;o 52 D
{ 7e4k,fP;daole]a[9eo;o 525 ; 133426 c]t 1112 453606 D

187

[qfmu @$ g7njv’\kp∈ c]t∉ c]t dj5,gmqjkdao

!D g7jnv’\kp ∈ c]t ∉

dyf9tdemu !

dD xtgmf]k;\kpfh;pA F dj5,vk-Po\kpfh;p A D xtgmf]k;vq’xtdv[0v’ d5j,
vk-Po lk,kf0Pogxaoxt3spdlaopt]adG A∈A

0D xtgmf9uo< c [+gxaovq’xtdv[0v’d5j, A <F 0Po laopt]ad"c∉A"D
7D P c,oj d5j,9eo;ome,t-kf7j6musj ^kpdj;k 3c]t |hvpd;j k13.sh0Pod5j, P c[[
0tspkp P = { 3,..................................,13}

la’gdfg[yj’xt3spd8+wxoU c]h;.sh\kp 4nd s^n zyf .lj[jvos;jk’

4∈P......................................... 14∉P.......................................

     5∈P......................................... 12∈P.......................................
5∉P......................................... 24∉P.......................................

0, 34∉P ................................... 16∉P .......................................ສະຫງສ.ວວນ.ລິສຂະສິດ

’D 9q’j la’gdfg[y’j I6[d5j, T c]t M TM
{ la’gdfg[y’j xt3spd8+wxoU c]h;

\kp 4nd s^n zyf .l[j jvo9Eg,af da h
e 2
4 m

     e∈ T .........................................  
4∈ T ......................................... e∈M .......................................
d ∈T ......................................... 4∈M .......................................
a∈ T ......................................... d ∈M .......................................
m∈T ......................................... a∈M .......................................
h∈ T ......................................... m∈M .......................................
2∈ T ......................................... h∈M .......................................
2∈M .......................................

188

{ .o8k8t]k’g[Nv’0;koU 9q’j \kp ! vq’xtdv[ T M
8k,c4; e c]t 4ao T le]a[ e∈T
9qj’\kp 0 8k,c4; e c]t 4ao M e1 0
le]a[ e∉M c]t xt8y[affaj’0hk’gmy’
le]a[vq’xtdv[8+Mwx. 4 ........ ........
d ........ ........

a1 1

m ........ ........

h0 1

2 ........ ........

9D { dj5, A ,uvq’xtdv[ 4 vao c]t dj5, B ,uvq’xtdv[3 vaoF Ih6;kj

, ∈ A; , ∉ B; 4 ∉ A; 4 ∈ B; i ∉ A; i ∈ B; m ∈ A; m ∈ B;

u ∈ A; u ∉ B; 0 ∈ A; 0 ∉ B

{ .sh8nj,.lj A = { ..........................}; B = { ..........................}
ສະຫງສ.ວວນ.ລິສຂະສິດ
{ .sh8jn,g]d 0 s^n 1.l8j k8t]k’0hk’0;koUD vq’xtdv[ A B
{ .sh8jn,vq’xtdv[0v’d5j, A c]t d5j, B
,1 0
rhv,ma’;kj ’vq’xtdv[0v’d5,j .sh4nd8hv’
.oI6[]5j,oUD 4 ........ ........
i ........ ........
AB ........
........
m ........ ........

i u ........
0 ........
...... ......

.97;k,

 ∈ c]t ∉ gxaog7njv’\kp7toyflkf7ndaoda[ +; −; =; c]t vjnoMmuoj adIPo
g7upIh6,kdvj oD

 g7njv’\kp ∈.-hcmo7e;jk gxaovq’xtdv[0v’D
 g7jnv’\kp∉.-hcmo7e;kj [+gxaovq’xtdv[0v’D

189

ສະຫງສ.ວວນ.ລິສຂະສິດ@D d5j,gmqjkdao

dyf9tdemu !
dD gIqklk,kf0PoG 3 + 7 = 10 S ltcf’;jk 3 + 7 c]t 10 c,jo9eo;ofP;dao
{ xt3spd 3 + 5 = 7 c,joxt3spd 4nd s^n zyf LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
{ gIqklk,kf0PoG 3 + 5 ≠ 7 gxaoxt3spdmuj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
0D [aofkd5j,8jk’M 8+wxoU]h;oc]h;c8cj ,jod5j,muj,uvq’xtdv[ 3vaoD 9qj’-vdskd5j,muj
,uvq’xtdv[7ndaoG

A = { ; 2; U}; B = { 2; U; 0 }; C = { U; 0; };D = { 0; 2; U};
E = { ; U; 0 }

8q;1jk’ !D

d5j, B c]t D ,uvq’xtdv[7ndao gryjog;qhk;jk dj5, B gmqjkda[d5j, D gIqkwfh B = D
ovd9kdoU ,udj5,.fcfjmujgmqkj dao L

8q;1jk’ @D

dj5, A da[d5j, B ,u9eo;ovq’xtdv[gmqjkdao[+ L
c8j  gxaovq’xtdv[0v’ A c8j .....................B c]t 0.....................B c8j 0
[+c,jovq’xtdv[0v’ A gryjog;qhk;jk dj5, A 8kj ’d5j, B gIqkwfh A ≠ B
{ ovd9kdoU ,udj5,.fvudmu8j kj ’9kdd5j, A L phvospa’ L
{ d5j, C dj5, D 8jk’dao[+ L

dyf9tdemu @

dD lv’dj5,8+wxoU { dF 0F7F 9} c]t { 7F0F 9} [+gmqjkdaophvospa’ LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

0D dj5, X = { 6,2,4 } c]t d5j, U 9eo;o76j]ts;kj ’1 c]t 7 d5j, X c]t U gmqjk
dao[+ L 9qj’vtmy[kpD

7D dj5, A = { dF fF 8F9F[F 1} c]t d5j, E c,jovadlvodk’.ortpao-tot]k; gmqjk

dao[+ L phvospa’ L
’D d5j, F = {1, 4, 3, 2 } c]t E c,jodj5,8q;g]d.-h0Po9eo;o 142332 gmqkj dao[+L

190

ສະຫງສ.ວວນ.ລິສຂະສິດ.97;k,

 m5dM vq’xtdv[0v’d5,j X c,oj vq’xtdv[0v’d5j, U c]t m5dMvq’xt
dv[0v’d5j, U c,jovq’xtdv[0v’dj5, X D gryjogvUo d5j, X c]t dj5, U
c,jodj5,vaofP;daoD gryoj 0Po X = U s^n U = X D

 4hk;kj lv’d5j, A c]t E [+c,oj dj5,fP;daoD gryjo0Po A ≠ E

#D [qfg/ydsaf

!D 9qj’8nj,g7njv’\kp = s^n ≠ .l.j oxt3spd]j5,oUG

{ a, b}.........{ b, a}; { a, b}.........{ c, b}; {1, 2, 3}.........{ 3, 2, 1};

{ 27, 52}.........{ 25, 72}; { a}.........{ a},{ a}.........{ b}.

@D .oltg\uzqo]5j,oU .shoadIPo8jn,.lj[vj o0v’vq’xtdv[musj kpwxD

{ 2, 3} = { 3,.......}; { a, b } = { ....,....}; { ....,....} = { 3, 2, 1};

{ u, v, w} = { w,....,....}

#D d5j, I = {1, 3, 5, 7, 9 } D oadIPozh6\b’j 0Podj5,c[[sph=;jkG I c,oj d5j,9eo;ome,t
-kf7ud oadIPo7yf;jk 4nd s^n zyf L phvospa’L
$D .shoadIPo0Podj5,8+wxouh gxaod5j,c[[sp=hG { 2, 4, 6, 8, 10 } S { 5, 13, 9, 7, 11} D
&D { M c,jodj5,0v’gfnvomu,j u 30;aoD 9q’j IPo0Pod5j, M c[[0tspkpG

M = { ,u45okFg,lkFDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD }
{ F = { ,a’dvoF ,uokF rnfltrkF d=]tdqfF ly’skFDDDDDDDDDDDDDDDDDDDD }
9q’j la’gdfvq’xtdv[0v’d5j, F c]h; 0Po8nj,.lj.sh7q[4h;o rhv,ma’de oqfdj5, F
c[[sp=hD

*D{ ` c,oj 9eo;ome,t-kfD 4k,;kj gIqklk,kf0Povq’xtdv[0v’dj5, ` c[[

0tspkpwfh\qfm5d8q;[+ L phvospa’ L
{ g-aoj fP;daoda[9eo;ome,t-kf7j6 d5j,oU[+lk,kf0Povq’xtdv[0v’d5j,c[[

0tspkpwfh\qf phvospa’ LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
{ g[yj’[qfg/ydsaf0=h 5 9tgsao;jk d5j, F s^n M lk,kf0Povq’xtdv[0v’d5j,wfh7q[
gIqkgvUod5j, F s^n M ;kj dj5,lYol5f phvo;kj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

191

(D gryjo.shd5j, A = { 2, 5, 13,8, 6, 7, 14, 4 } AB
...... ...... ......
s^n dj5, B = {1, 3, 4,7, 6, 5, 12, 13}

{ 9qj’8nj,vq’xtdv[0v’dj5, A s^n 0v’
dj5, B .l.j oI6[d5j,mu1j j60hk’0;koUD

{ d6j, A c]t B gmqkj dao s^n [+ L

)D gryoj .shI6[cxfd5j,fa’j oUG  

C  l  E  t  v  u  G 
p mn
a  A f  g r  s w
B  h  i q x  y  z  
b  c j k  D  H 
d  F 

e 

{ 9qj’\kp 4nd s^n zyf .ljxt3spd8+wxoUG ສະຫງສ.ວວນ.ລິສຂະສິດ

A = B................; C = D................; E = F................;G = H................

{ 8q;1kj ’ gryjogvqkvq’dv[ A vvd9kdd5j, E c]t p,g, w vvd9kdd5j,  F grjnv.sh

dj5,.\j E′ = F′D

grnjv.shd5j,.\j C′ = D′ F oadIPo7;ogvqkvq’xtdv[.fvvd9kddj5,  C c]t  D L

grjnv.shd5j,.\j A′ = B′ F oadIPo7;ogvqkvq’xtdv[.fvvd9kddj5,  A c]t  B L

grjnv.shd5j,.\j G′ = H′F oadIPo7;ogvqkvq’xtdv[.fvvd9kddj5, G c]t  H L

dj5,.\j G′ = H′ gvUo;jk dj5,spa’ L a w  A 
_D 9q’j la’gdfg[yj’I6[dj5,0hk’0;koU D b  c  f 
dD 9q’j 8n,j .lj[vj o9Eg,af
{ vq’xtdv[0v’dj5, A ,u7n DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD e 

{ vq’xtdv[0v’d5j, B ,u7n DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD d  k  B 

0D oadIPo8hv’gvqkvq’xtdv[8q;.fcfjvvd9kdd5j, A c]t vvd9kddj5, B grnjv.sh
d5j,.\j  A′ = B′

!WD d5j, {a,b,c,d} c]t d5j, {c,b,d} [+gmqkj dao phvospa’ L
dj5, {c,b,d} c]t d5j, {5,b,d} [+gmqjkdao phvospa’ L

192

 

[qfmu @& vto5dj5, c]t czo;kf0v’dj5,

!D vto5d5j, Xs^n rkdlj;o0v’dj5,? B 
A 
dyf9tde
dD .shoadIPola’gdfg[yj’I6[d5j, A c]t B b  a 
{ a c,oj vq’xtdv[0v’dj5, B S a d=c,oj vq’xtdv[ c  m  d 
0v’dj5, A g-aoj fP;daoD
{ m c,joDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDd5j, B S m d=c,jo n  u 
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDdj5, A g-aoj fP;daoD
{ d ∈ B; d .................A g-ajofP;daoD M   P ສະຫງສ.ວວນ.ລິສຂະສິດ
{ m5dMvq’xtdv[0v’ B c,oj vq’xtdv[0v’ A
c,oj [+ L 1  8  
{ gryjogvUo;kj G dj5, B c,jogxaorkdl;j oobj’0v’
dj5, A s^n gvUo;kj G B gxaovto5d5j,0v’d5j, A 4 
{ d6j, { c, b, u }c,jo vto5dj5,0v’d5j, A [+ L 2 
phvospa’ L 6  5 
{ dj6,{ c, u, h } [+c,oj vto5d5j,0v’d5j, A phvo 7  9 
spa’L 3 
0D .sh0Podj5, M,P, S, T c[[0tspkp
S  T 
M = {...........}; P = {............}

S = { ..........}; T = { ...........}

{ P gxaovto5dj5,0v’ M [+ L
{ T gxaovto5d5j,0v’ P [+ L
{ T gxaovto5dj5,0v’S [+ L phvospa’ L

193

 

@D .97;k,

d5j, B c,jovto5d5,j 0v’ A d=8+g,nvj ;jk \qfm5dMvq’xtdv[0v’ B c,oj gxaovq’
xtdv[0v’ A D gIqkwfh B c,jorkdlj;o0v’ A D
gryjo0Pofa’j oUG B c,oj vto5dj5,0v’ A

s^n B c,jorkdl;j o0v’d5j, A 0Pogxao B ⊂ A D

\kpgsfG
g7njv’\kp⊂ S E ⊂ F gIqkwfh E c,oj vto5d5,j 0v’ F

g7nvj ’\kp∈S e ∈ F gIqkwfh e c,jovq’xtdv[0v’ F

czo;kfI6[d5j,

gryoj .shd5j, E F A c,oj vto5dj5,0v’ E c]t B c,oj rkdlj;o0v’ E fa’j oUD

E = { 3, , 7, a,U, 0, 2, x };A = { , a,3,U};B = { 0, , U, x,3}

 A E dD .oI6[mu!ma’lv’c[[.sh ສະຫງສ.ວວນ.ລິສຂະສິດAE
gvqkvq’xtdv[0v’ A
a  7
.......... x c]t0v’ E .lj.oma’
lv’I6[.sh7q[D
a 72

E 0D .oI6[mu@ma’lv’c[[.sh E
gvqkvq’xtdv[0v’ B
B ............. c]t 0v’ E .l.j oma’ 7 ...
lv’I6[.sh7q[D
...........7 B ....

7D3I,gvqklv’I6[g[Nv’ AE
A E gmy’c]h;gvqkvq’xtdv[

B ... 0v’ A F B c]t E .lj.oma’ a .,... 0

7 ... lv’I6[.sh7q[4h;o B 7

Xczo;kf0v’ Carroll ? Xczo;kf0v’ Venn ?

Lewis Carroll ( 1832 − 1898 ) c]t John Venn ( 1834 − 1923 ) ma’lv’7qo

gxaooad7toyflkf-k;va’dyfD

194

 

#D [qfg/ydsaf

!D ` c,jod5j,9eo;ome,t-kf
W c,jodj5,0v’;ao.o|b’j vkmyf
L c,jodj5,0v’rtpao-tot]k;

9qj’8nj,.ljxt3spd8+wxoU3fp.sh.-hg7jnv’\kp∈s^n ∉s^n ⊂ D

5............` S 0,72............`
S
{ 1 }..........` d..........L

a ............L S ;aor5f ...........w

{1, 2, 3, 4 }.............` S 1............`

-6............` S { dF 0F 7 }..........L
{d}............L
;aova’7ko...........L S

@D E = { , U,1} xt3spd8+wxoU 4nd s^n zyf E ⊂ E............................

gIqkg;qhkwfh;kj G E ............................ E gv’ ສະຫງສ.ວວນ.ລິສຂະສິດ

#D A c,oj d5j,9eo;ome,t-kf|hvpd;j k13

B c,jodj5,9eo;omt;u76o0v'3 muj|hvpdj;k13

C c,oj dj5,9eo;omt;u76o0v’ 6 muj|hvpd;j k13

 .sh0Pod5j, A,B,C c[[0tspkp

A = { 0,1, 2,......................................................13 }

B = { 0,3,..............................................................}

C = { 0, 6,..............................................................}

vto5d5j,0v’ A ,udj5,.fcfj L c]t vto5d5,j 0v’ B ,ud5,j .fcfj L

$D .shoadIPola’gdfg[yj’I6[dj5,]j5,oUG

E E = R ⎧R ⊂ E R c, oj vtod5 5j,0v'E c]t
⎨⎩E ⊂ R E c, oj vtod5 5j,0v'R

R Â R c,jolj;og8a,0v’dj5, E

V Â V c,oj l;j ogxqkj 0v’d5j, E

.sh\kp 4nd s^n zyf .ljxt3spd8+wxoUG

195

ສະຫງສ.ວວນ.ລິສຂະສິດ 

E ⊂ E........................... ; φ ⊂ E.............................

&D I6[dj5,8+wxoU c,oj 8q;cmo0v’dj5, A c]t B

AB

 4hk;jk A ⊂ B F .shc8h,7no.\3j fp.sh0uf0hkle]a[rNomuj “ gxqjk ”
 4hk;kj B ⊂ A F .shc8h,7no.\3j fp.sh0uf0hkle]a[rNomuj “ gxqjk ”
 4hk;jk A = BF .shc8h,7no.\j3fp.sh0j uf0hkle]a[rNomuj “ gxjqk ”
*D gryjo.shdj5, J = { ;ao9aoF ;aor5fF ;aol5dF ;ao va’7koF;aortsaf}

S = c,jod5j,0v’ ,N.o|b’j vkmyfD
dD .shoadIPodeoqfdj5, J c[[sp=hD
0D d5j, S c]t dj5, J gmqjkdao[+ L 4hk[+gmqjkdao9tgvqkvq’xtdv[.f8nj,.ljdj5, J grjnv
.shgmqkj da[d5j, S L
(D gryoj gvUo V c,oj dj5,vadlvol6’sqd8q;F W c,oj d5j,rtpao-totluj8q;7nG dF7F’F9D
dD { .shc8h,czo;kf0v’Venn d5j, A muj,urtpao-totly[8q;0hk’gmy’D

{ m5dMvq’xtdv[ V c,jogxaovq’xtdv[0v’ A [+ L gIqklk,kfg;qhkwfhc[[.f]t
s;kj ’dj5, V c]t dj5, A L s^n0Po V.........A
{ m5dMvq’xtdv[0v’ W c,oj vq’xtdv[0v’ A [+ L gIqklk,kfg;qhkwfhc[[.f
]ts;kj ’d5j, W c]td5j, A L s^n 0Po W............A
0D .oczo;kf0v’d5,j A .sh\kpI6[0v’d5j, V c]t W D .sh;k’rtpao-tot
3 8q;7nG fF-F8D .sh4nd[jvoD
)D gryjogvUo;kj d5j, A gxaod5j,9eo;o-bj’ A = { 6,14,2,8,10 } D
P c,jodj5,9eo;ome,t -kf76j|hvpdj;k16F E c,jod5j,9eo;ome,t-kf|hvp
dj;k16D gryoj lk,kf0Powfhco;.f]ts;jk’d5j, A c]t d5j, P S A c]t E S
P c]t E L
_D dD gryoj .shc8h,czo;kf0v’ venn le]a[d5j,]5,j oUG
` c,jodj5,9eo;ome,t-kf F I c,jod5j,9eo;ome,t-kf7ud c]t D c,jo
d5j,9eo;ome,t-kf-b’j sq;|j;pc,oj 5D

196