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સંબંધ અને વિધેય

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ગણિત ધોરણ 12 સેમિસ્ટર 3

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નીચે આપેલ સંબંધ સ્વવાચક, સંમિત કે પરંપરિત છે ?
A =Z, S = {(x, y) | x - y bold element of Z}


નીચે આપેલ સંબંધ સ્વવાચક, સંમિત કે પરંપરિત છે ?
A = N, S = {(x, y) | x એ y વડે વિભાજ્ય હોય}


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નીચે આપેલ સંબંધ સ્વવાચક, સંમિત કે પરંપરિત છે ?
A = {1, 2, 3, ...., 10}, S = {(x, y) | y = 2x}


અહીં, y = 2x તથા bold x bold comma bold space bold y bold space bold element of bold space bold A bold space open parentheses bold because bold space bold S bold space bold subset of bold space bold A bold space bold cross times bold space bold A close parentheses હોવાથી,



bold x bold space bold equals bold space bold 1 bold space bold element of bold space bold A bold space bold મ ા ટ ે bold comma bold space bold y bold space bold equals bold space bold 2 bold space open parentheses bold 1 close parentheses bold space bold equals bold space bold 2 bold space bold element of bold space bold A

bold therefore bold space bold S presubscript bold 1 subscript bold 2 bold space bold space bold space bold therefore bold space open parentheses bold 1 bold comma bold space bold 2 close parentheses bold space bold element of bold space bold S


bold x bold space bold equals bold space bold 2 bold space bold element of bold space bold A bold space bold મ ા ટ ે bold comma bold space bold y bold space bold equals bold space bold 2 bold space open parentheses bold 2 close parentheses bold space bold equals bold space bold 4 bold space bold element of bold space bold A

bold therefore bold S presubscript bold 2 subscript bold 4 bold space bold space bold space bold therefore bold space open parentheses bold 2 bold comma bold space bold 4 close parentheses bold space bold element of bold space bold S


bold x bold space bold equals bold space bold 3 bold space bold element of bold space bold A bold space bold મ ા ટ ે bold comma bold space bold y bold space bold equals bold space bold 2 bold space open parentheses bold 3 close parentheses bold space bold equals bold space bold 6 bold space bold element of bold space bold A

bold therefore bold space bold S presubscript bold 3 subscript bold 6 bold space bold space bold space bold therefore bold space open parentheses bold 3 bold comma bold space bold 6 close parentheses bold space bold element of bold space bold S


bold x bold space bold equals bold space bold 4 bold space bold element of bold space bold A bold space bold મ ા ટ ે bold comma bold space bold y bold space bold equals bold space bold 2 bold space open parentheses bold 4 close parentheses bold space bold equals bold space bold 8 bold space bold element of bold space bold A

bold therefore bold space bold S presubscript bold 4 subscript bold 8 bold space bold space bold therefore bold space open parentheses bold 4 bold comma bold space bold 8 close parentheses bold space bold element of bold space bold S


bold x bold space bold equals bold space bold 5 bold space bold element of bold space bold A bold space bold મ ા ટ ે bold comma bold space bold y bold space bold equals bold space bold 2 bold space open parentheses bold 5 close parentheses bold space bold equals bold space bold 10 bold space bold element of bold space bold A

bold therefore bold space bold S presubscript bold 5 subscript bold 10 bold space bold space bold therefore bold space open parentheses bold 5 bold comma bold space bold 10 close parentheses bold space bold element of bold space bold S

bold પર ં ત ુ bold space bold x bold space bold equals bold space bold 6 bold comma bold space bold 7 bold comma bold space bold 8 bold comma bold space bold 9 bold comma bold space bold 10 bold space bold element of bold space bold A bold space bold મ ા ટ ે bold space bold y bold space bold not an element of bold space bold A

bold therefore bold space bold S bold space bold equals bold space open curly brackets open parentheses bold 1 bold comma bold space bold 2 close parentheses bold comma bold space open parentheses bold 2 bold comma bold space bold 4 close parentheses bold comma bold space open parentheses bold 3 bold comma bold space bold 6 close parentheses bold comma bold space open parentheses bold 4 bold comma bold space bold 8 close parentheses bold comma bold space open parentheses bold 5 bold comma bold space bold 10 close parentheses close curly brackets


(1) સ્વવાચકતા માટે,

bold 1 bold space bold element of bold space bold A માટે bold S presubscript bold 1 subscript bold 1 એટલે કે, open parentheses bold 1 bold comma bold space bold 1 close parentheses bold space bold not an element of bold space bold S

આમ, Aમાં ઓછામાં ઓછો એક ઘટક 1 એવો મળે છે કે જેથી bold S presubscript bold 1 subscript bold 1 શક્ય નથી.

S એ સ્વવાચક સંબંધ નથી.

(2) સંમિતતા માટે,

અહીં, bold S presubscript bold 1 subscript bold 2  છે પરંતુ bold S presubscript bold 2 subscript bold 1 અસત્ય છે.

એટલે કે, open parentheses bold 1 bold comma bold space bold 2 close parentheses bold space bold element of bold space bold S  છે પરંતુ open parentheses bold 2 bold comma bold space bold 1 close parentheses bold space bold not an element of bold space bold S

ઓછામાં ઓછો એક ઘટક open parentheses bold 1 bold comma bold space bold 2 close parentheses bold space bold element of bold space bold S મળે છે કે જેથી open parentheses bold 2 bold comma bold space bold 1 close parentheses bold space bold not an element of bold space bold S

S સંમિત સંબંધ નથી.

(3) પરંપરિતતા માટે,

અહીં, bold S presubscript bold 1 subscript bold 2 bold space bold અન ે bold space bold S presubscript bold 2 subscript bold 4 છે એટલે open parentheses bold 1 bold comma bold space bold 2 close parentheses bold space bold element of bold space bold S bold space bold અન ે bold space open parentheses bold 2 bold comma bold space bold 4 close parentheses bold space bold element of bold space bold S પરંતુ open parentheses bold 1 bold comma bold space bold 4 close parentheses bold space bold not an element of bold space bold S

S પરંપરિત સંબંધ નથી.


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નીચે આપેલ સંબંધ સ્વવાચક, સંમિત કે પરંપરિત છે ?
A = {1, 2, 3, 4, 5, 6}, , S = {(x, y) | x એ y વડે વિભાજ્ય હોય}


નીચે આપેલ સંબંધ સ્વવાચક, સંમિત કે પરંપરિત છે ?
A =R, S = {(x, y) | y = x + 1}


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