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ગણિત ધોરણ 9 પ્રથમ સત્ર

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ગણિત

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જો bold A bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold Z bold comma bold space bold x to the power of bold 2 bold space bold minus bold space bold x bold space bold equals bold space bold 0 close curly brackets bold comma bold space bold B bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold 1 bold space bold less-than or slanted equal to bold space bold x bold space bold less-than or slanted equal to bold 4 close curly brackets,  તો bold A bold space bold subset of bold space bold B કહી શકાય ? શા માટે ?


bold A bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold Z bold comma bold space bold x to the power of bold 2 bold space bold minus bold space bold x bold space bold equals bold space bold 0 close curly brackets

bold x to the power of bold 2 bold space bold minus bold space bold x bold space bold equals bold space bold 0 bold space bold space bold space bold space bold therefore bold space bold x bold space open parentheses bold x bold space bold minus bold space bold 1 close parentheses bold space bold equals bold space bold 0

bold therefore bold space bold x bold space bold equals bold space bold 0 bold space bold space bold space bold અથવ ા bold space bold space bold x bold space bold equals bold space bold 1 bold space bold equals bold space bold 0

bold therefore bold space bold x bold space bold equals bold space bold 0 bold space bold space bold space bold અથવ ા bold space bold space bold x bold space bold equals bold space bold 1



હવે, bold 0 bold space bold element of bold space bold Z bold space bold space bold તથ ા bold space bold space bold 1 bold space bold element of bold space bold Z

ગણ A = {0, 1}

હવે, bold B bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold 1 bold space bold less-than or slanted equal to bold space bold x bold space bold less-than or slanted equal to bold space bold 4 close curly brackets

અહીં, bold x bold space bold element of bold space bold N bold space bold space bold તથ ા bold space bold space bold 1 bold space bold less-than or slanted equal to bold space bold x bold space bold less-than or slanted equal to bold space bold x  છે.

x = 1, 2, 3, 4

ગણ B = {1, 2, 3, 4}

અહીં, bold 0 bold space bold element of bold space bold A bold space bold semicolon bold space bold પર ં ત ુ bold space bold 0 bold space bold not an element of bold space bold B

આમ, ગણ Aનો પ્રત્યેક ઘટક ગણ Bનો ઘટક નથી.

ગણ A એ ગણ Bનો ઉપગણ નથી.

 B

bold therefore bold space bold A bold space bold subset of bold space bold B એમ કહી શકાય નહિ.


જો U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} , A = {1, 2, 4, 6, 8}, તો A’ શોધો અને બતાવો કે bold A bold space bold union bold space bold A bold apostrophe bold space bold equals bold space bold U bold.


U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} અને A = {1, 2, 4, 6, 8} 

હવે, bold A bold apostrophe bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold U bold comma bold space bold x bold space bold not an element of bold space bold A close curly brackets

એટલે કે Uમાં હોય, પરંતુ ગણ Aમાં ન હોય તેવા ઘટકો 3, 5, 7, 9 અને 10 છે.

A’ = {3, 5, 7, 9, 10}

હવે,

bold A bold space bold union bold space bold A bold apostrophe bold space bold equals bold space bold left curly bracket bold 1 bold comma bold space bold 2 bold comma bold space bold 4 bold comma bold space bold 6 bold comma bold space bold 8 bold right curly bracket bold space bold union bold space bold left curly bracket bold 3 bold comma bold space bold 5 bold comma bold space bold 7 bold comma bold space bold 9 bold comma bold space bold 10 bold right curly bracket

bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold equals bold space bold left curly bracket bold 1 bold comma bold space bold 2 bold comma bold space bold 3 bold comma bold space bold 4 bold comma bold space bold 5 bold comma 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 right curly bracket bold space

bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold equals bold space bold U



bold therefore bold space bold A bold space bold union bold space bold A bold apostrophe bold space bold equals bold space bold U સાબિત થાય છે.


નીચેર આપેલ ગણને ખાલી ગણ કે એકાકી ગણમાં વર્ગીકૃત કરો :
open parentheses bold 1 close parentheses bold space bold A bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold Z bold comma bold space bold x bold space bold plus bold space bold 1 bold space bold equals bold space bold 0 close curly brackets

open parentheses bold 2 close parentheses bold space bold B bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x to the power of bold 2 bold space bold minus bold space bold 1 bold space bold equals bold space bold 0 close curly brackets

open parentheses bold 3 close parentheses bold space bold C bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x bold space bold એ bold space bold 13 bold space bold અન ે bold space bold 17 bold space bold વચ ્ ચ ે ન ી bold space bold અવ િ ભ ા જ ્ ય bold space bold સ ં ખ ્ ય ા close curly brackets


open parentheses bold 1 close parentheses bold space bold A bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold Z bold comma bold space bold x bold space bold plus bold space bold 1 bold space bold equals bold space bold 0 close curly brackets

અહીં, x + 1 = 0     x = -1 અને negative 1 space element of space straight Z છે.      ગણ A = -1

ગણ A માં એક જ સભ્ય છે માટે તે એકાકી ગણ છે.


open parentheses bold 2 close parentheses bold space bold B bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x to the power of bold 2 bold space bold minus bold space bold 1 bold space bold equals bold space bold 0 close curly brackets

bold અહ ીં bold comma bold space bold x to the power of bold 2 bold space bold minus bold space bold 1 bold space bold equals bold space bold 0 bold space bold space bold space bold space bold space bold therefore bold space bold x to the power of bold 2 bold space bold equals bold space bold 1 bold space bold space bold space bold therefore bold space bold x bold space bold equals bold space bold 1 bold space bold અથવ ા bold space bold x bold space bold equals bold space bold minus bold 1 bold space bold space bold space bold space bold space bold space open parentheses bold because bold space bold 1 to the power of bold 2 bold space bold equals bold space bold 1 bold space bold અન ે bold space open parentheses bold minus bold 1 close parentheses to the power of bold 2 bold space bold equals bold space bold 1 close parentheses


વળી, bold 1 bold space bold element of bold space bold N છે પણ bold minus bold 1 bold space bold not an element of bold space bold N    ગણ B = {1}, ગણ B માં એક જ સભ્ય છે માટે તે એકાકી ગણ છે.


open parentheses bold 3 close parentheses bold space bold C bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x bold space bold એ bold space bold 13 bold space bold અન ે bold space bold 17 bold space bold વચ ્ ચ ે ન ી bold space bold અવ િ ભ ા જ ્ ય bold space bold સ ં ખ ્ ય ા close curly brackets


અહીં, 13 અને 17 ની વચ્ચે bold 14 bold space bold equals bold space bold 7 bold space bold cross times bold space bold 2 bold comma bold space bold 15 bold space bold equals bold space bold 5 bold space bold cross times bold space bold 3 bold comma bold space bold 16 bold space bold equals bold space bold 4 bold space bold cross times bold space bold 4 bold spaceએ તમામ વિભાજ્ય સંખ્યાઓ છે, એટલે કે એક પણ સંખ્યા અવિભાજ્ય નથી.

ગણ C એ ખાલી ગણ છે.

નીચે આપેલ ગણને સમાન ગણ અથવા સામ્ય ગણમાં વર્ગીકૃત કરો :
open parentheses bold 1 close parentheses bold space bold A bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x bold space bold less-than or slanted equal to bold space bold 7 close curly brackets

bold space bold space bold space bold space bold space bold B bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold Z bold comma bold space bold minus bold 3 bold space bold less-than or slanted equal to bold space bold x bold space bold less-than or slanted equal to bold space bold 3 close curly brackets

open parentheses bold 2 close parentheses bold space bold A bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x bold space bold એ bold space bold 2 bold space bold ન ો bold space bold ગ ુ ણક bold space bold છ ે bold comma bold space bold x bold space bold less than bold space bold 10 close curly brackets

bold space bold space bold space bold space bold space bold B bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x bold space bold એ bold space bold 10 bold થ ી bold space bold ન ા ન ી bold space bold ય ુ ગ ્ મ bold space bold પ ્ ર ા ક ૃ ત િ ક bold space bold સ ં ખ ્ ય ા bold space bold છ ે bold. close curly brackets


open parentheses bold 1 close parentheses bold space bold A bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x bold space bold less-than or slanted equal to bold space bold 7 close curly brackets

bold therefore bold space bold italic A bold space bold equals bold space bold left curly bracket bold 1 bold comma bold space bold 2 bold comma bold space bold 3 bold comma bold space bold 4 bold comma bold space bold 5 bold comma bold space bold 6 bold comma bold space bold 7 bold right curly bracket




bold B bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold Z bold comma bold space bold minus bold 3 bold space bold less-than or slanted equal to bold space bold x bold space bold less-than or slanted equal to bold space bold 3 close curly brackets

bold therefore bold space bold B bold space bold equals bold space open curly brackets bold minus bold 3 bold comma bold space bold minus bold 2 bold comma bold space bold minus bold 1 bold comma bold space bold 0 bold comma bold space bold 1 bold comma bold space bold 2 bold comma bold space bold 3 close curly brackets


અહીં, bold 7 bold space bold element of bold space bold A પરંતુ bold 7 bold space bold not an element of bold space bold B હોવાથી, ગણ A અને ગણ Bના ઘટકો સમાન નથી.

ગણ A અને ગણ B સમાન ગણ નથી.

અહીં n(A) = 7 અને n(B) = 7 . આથી ગણ A અને ગણ Bમાં ઘટકોની સંખ્યા સરખી છે.

ગણ A અને ગણ B સામ્ય ગણ છે.




open parentheses bold 2 close parentheses bold space bold A bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x bold space bold એ bold space bold 2 bold space bold ન ો bold space bold ગ ુ ણક bold space bold છ ે bold comma bold space bold x bold space bold less than bold space bold 10 close curly brackets

પ્રાકૃતિક સંખ્યા ગણ Nના 10થી નાના 2 ના ગુણક 2, 4, 6, 8 છે.   A = {2, 4, 6, 8}

bold B bold space bold equals bold space open curly brackets bold x bold divided by bold x bold space bold element of bold space bold N bold comma bold space bold x bold space bold એ bold space bold 10 bold space bold થ ી bold space bold ન ા ન ી bold space bold ય ુ ગ ્ મ bold space bold પ ્ ર ા ક ૃ ત િ ક bold space bold સ ં ખ ્ ય ા bold space bold છ ે bold. close curly brackets

પ્રાકૃતિક સંખ્યા ગણ Nની 10થી નાની યુગ્મ પ્રાકૃતિક સંખ્યાઓ 2, 4, 6, 8 છે.   B = {2, 4, 6, 8}

આમ, A = {2, 4, 6, 8} અને B = {2, 4, 6, 8}

અહીં ગણ A અને ગણ Bના બધા ઘટકો સમાન છે.

ગણ A અને ગણ B સમાન ગણ છે.


ગણ A = {1, 2, 3}ના ઉપગણોની સંખ્યા શોધો તથા ગણ Aના તમામ ઉપગણો લખો.


ગણ A = {1, 2, 3}

અહીં ગણ Aની સંખ્ય-સંખ્યા 3 છે. n = 3

હવે, ગણ A ના ઉપગણોની સંખ્યા

bold equals bold space bold 2 to the power of bold n

bold equals bold space bold 2 to the power of bold 3

bold equals bold space bold 2 bold space bold cross times bold space bold 2 bold space bold cross times bold space bold 2

bold equals bold space bold 8


ગણ Aના તમામ ઉપગણો

bold empty set bold comma bold space open curly brackets bold 1 close curly brackets bold comma bold space open curly brackets bold 2 close curly brackets bold comma bold space open curly brackets bold 3 close curly brackets bold comma bold space open curly brackets bold 1 bold comma bold space bold 2 close curly brackets bold comma bold space open curly brackets bold 1 bold comma bold space bold 3 close curly brackets bold comma bold space open curly brackets bold 2 bold comma bold space bold 3 close curly brackets bold comma bold space open curly brackets bold 1 bold comma bold space bold 2 bold comma bold space bold 3 close curly brackets