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à¤Ã×èͧÁ×ͧ͢ËÑÇ¢éÍ | ¤é¹ËÒã¹ËÑÇ¢é͹Õé |
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Group ¾Õª¤³ÔµªèÇÂáÊ´§ÇÔ¸Õ·Ó·Õ¤ÃѺ
Let Z be that set of integers. Define operation \oplus on Z by a\oplus b = a+b-2 \forall a,b\in Z.
Show that (Z,\oplus ) is a group. 23 ÊÔ§ËÒ¤Á 2014 08:13 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Noker |
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ªèÇÂ˹è͹ФÃѺ ¼Á¾Öè§ÊÁѤäÃÑé§áá ãªé latex ÂѧäÁèà»ç¹ áµèá¹Ð¹Ó˹èͤÃѺ
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#3
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¨ÐáÊ´§¡ÒÃà»ç¹ group ¨ÐµéͧáÊ´§ÍÐäúéÒ§¤ÃѺ
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#4
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1. µéͧà»ç¹ a binary operation
2.µéͧÁÕ associative (semigroup) 3.ÁÕ identity 4.ÁÕ inverse ¹Õè¨Ð¶ÒÁ¤×Í ¨Ð¾ÔÊÙµÃÂѧ䧵ÒÁ⨷ÂìãËéà»ç¹ä»µÒÁà§×è͹䢹Õé¤ÃѺ à¾×èÍáÊà§ÇèÒÁѹà»ç¹ Group |
#5
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ÊÁºÑµÔ associative ¹ÕèÁѹà»ç¹Âѧä§àËÃͤÃѺ
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ÊÁºÑµÔ¡ÒÃÊÅѺ·Õè ¹èФÃѺ àªè¹ a(bc)=(ab)c
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#7
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¶éÒ§Ñ鹨оÔÊÙ¨¹ìä´éÁÑéÂÇèÒ
$a\oplus (b\oplus c) = (a\oplus b) \oplus c$
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¡è͹Í×蹹ФÃѺ µéͧàÃÔèÁ¨Ò¡¡ÒÃàªç¡¤Ø³ÊÁºÑµÔ 5 ¢éͤÃѺ
1.µéͧµÃǨÇèÒ $\mathbb{Z} $ äÁèãªè૵ÇèÒ§¤ÃѺ Íѹ¹Õé obviously ¤ÃѺ 2. µéͧµÃǨÊͺÇèÒ ÁÕÊÁºÑµÔ»Ô´ÀÒÂãµé $\oplus$ ÊÁºÑµÔ»Ô´¤×Í $\forall$ a,b $\in$ $\mathbb{Z} $ áÅéǵéͧáÊ´§ÇèÒ a$\oplus$ b $\in$ $\mathbb{Z} $§Ñº àÃÔèÁ¹Ð¤ÃѺ ãËé a,b $\in$ $\mathbb{Z} $ a$\oplus$ b = a+b-2 $\in$ $\mathbb{Z} $ ($\because $ $\forall$ a,b $\in$ $\mathbb{Z} $ ) $\therefore $ a$\oplus$ b $\in$ $\mathbb{Z} $ ´Ñ§¹Ñé¹ $\mathbb{Z} $ ÁÕÊÁºÑµÔ»Ô´ÀÒÂãµé $\oplus$ àÃÔèÁ§èÒÂáÅéÇãªèÁÑé¤ÃѺ 3.µéͧµÃǨÊͺÇèÒ $\forall$ a,b,c $\in$ $\mathbb{Z} $ (a$\oplus$b)$\oplus$c = a$\oplus$(b$\oplus$c) ¨Ðä´éÇèÒ (a$\oplus$b)$\oplus$c = (a+b-2)$\oplus$c = (a+b-2)+c-2 = a+b+c-4 áÅÐ a$\oplus$(b$\oplus$c) = a$\oplus$(b+c-2) = a +(b+c-2) -2 = a+b+c-4 ¾ºÇèÒ (a$\oplus$b)$\oplus$c = a$\oplus$(b$\oplus$c) ´Ñ§¹Ñé¹ $\oplus$ ÁÕÊÁºÑµÔ¡ÒÃà»ÅÕ蹡ÅØèÁ àËçÂÁÑé¤ÃѺÇèÒ 2 ¢é͹Õé§èÒÂæ 4.µéͧ¡ÒÃËÒàÍ¡Åѡɳì (àÃÒÅͧ·´ÊÔÇèÒ a$\oplus$ b = a áÅéÇ b ¨Ðà»ç¹ÍÐäà Åͧà»ÅÕè¹ a$\oplus$ b = a+b-2 ¨Ðä´éÇèÒ a+b-2 = a $\rightarrow $ b-2=0 ´Ñ§¹Ñé¹ b =2 ¹Ñè¹àͧ áÊ´§ÇèÒ 2 à»ç¹àÍ¡Åѡɳì ÍÂèÒÅ×Á¹Ð ¹Õè·´ã¹ã¨ 555) $\exists $2 $\in$ $\mathbb{Z} $ $\forall$ a $\in$ $\mathbb{Z} $ ·Õè·ÓãËé a$\oplus$ 2 = a+2-2 = a áÅÐ 2$\oplus$ a = 2+a-2 = a $\therefore $ a$\oplus$ 2 = a = 2$\oplus$ a ´Ñ§¹Ñé¹ 2 à»ç¹àÍ¡ÅѡɳìÀÒÂãµé $\oplus$ ¹Ñè¹á¹èÐÍÕ¡¹Ô´à´ÕÂǤÃѺ ¢éÍÊØ´·éÒ¹ÕèâË´ËÔ¹¹Ô´¹Ø§ 5. µéͧ¡ÒÃËÒ inverse (·´á»» ÇèÒ a$\oplus$b = 2 áÅéÇ b ¤×ÍÍÐäà ·Õèµéͧà»ç¹ = 2 µÒÁ¹ÔÂÒÁ¹Ð¤ÃѺ à¾ÃÒÐ 2 à»ç¹àÍ¡ÅÑ¡É³ì ·´¡è͹ àÃÒÃÙéÇèÒ a$\oplus$ b = a+b-2 ´Ñ§¹Ñé¹ a$\oplus$ b = 2 $\rightarrow $ a+b-2 = 2 ÂéÒ¢éÒ§ËÒ b ¨Ðä´éÇèÒ b = 4 - a ·´àÊÃç¨áÅéÇ) ãËé $\forall$ a $\in$ $\mathbb{Z} $ $\exists $4-a $\in$ $\mathbb{Z} $ (à¾ÃÒÐ a $\in$ $\mathbb{Z} $ áÅÐ 4 $\in$ $\mathbb{Z} $ ¨Ò¡ÊÁºÑµÔ»Ô´ÀÒÂãµé¡Òúǡ¢Í§¨Ó¹Ç¹¨ÃÔ§ ´Ñ§¹Ñé¹ 4-a $\in$ $\mathbb{Z} $ ) ¨Ðä´éÇèÒ a$\oplus$ (4 - a) = a+(4-a)-2 = 2 áÅÐ (4-a)$\oplus$ a = (4-a)+a-2 = 2 $\therefore $ a$\oplus$ (4 - a) = 2 = (4-a)$\oplus$ a ´Ñ§¹Ñé¹ 4-a à»ç¹µÑǼ¡¼Ñ¹¢Í§ a ÀÒÂãµé¡ÒôÓà¹Ô¹¡Òà $\oplus$ ¨Ò¡·Ñé§ 5 ¢é֧ͨÊÃØ»ä´éÇèÒ ($\mathbb{Z} $ ,$\oplus$) à»ç¹ ¡ÃØ» §Ñº ¨Ò¡·Õè·ÓÁÒäÁèÂÒ¡ãªèÁÑé¤ÃѺ ¶éÒʹ㨨зÓà¾ÔèÁ Âִ⨷Âìà´ÔÁ§Ñº(à§×è͹ä¢) áÅéÇËÒ a$\bullet $b = $\frac{ab}{2}$ ¨§áÊ´§ÇèÒ ($\mathbb{Z} $ ,$\bullet $ )à»ç¹ ¡ÃØ» àÍÒ㨪èǧѺ |
#10
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Íѹ¹Õéà»ÅÕ蹡ÅØèÁ¤ÃѺ áËÐæ
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#11
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à»ç¹ group ¨ÃÔ§ÃÖ
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#12
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µéͧ·Óº¹ ¨Ó¹Ç¹¨ÃÔ§¤ÃѺ ¢Íº¤Ø³¤ÃѺ 55555 ·Óº¹ Z äÁèÁÕÍÔ¹àÇÍÃìÊ ¤ÃѺ
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#13
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á¹è㨹Рä´éÅͧàªç¤¤Ãº·Ø¡¢éÍËÃ×ÍÂѧ
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ªèÇ·ӡÒúéÒ¹ group ˹èͤÃéÒÒ | pormath | ¤³ÔµÈÒʵÃìÍØ´ÁÈÖ¡ÉÒ | 2 | 03 ¾ÄÉÀÒ¤Á 2014 20:59 |
ªèÇ·ӡÒúéÒ¹ group ˹è͹ФР| pormath | ¤³ÔµÈÒʵÃìÍØ´ÁÈÖ¡ÉÒ | 1 | 04 àÁÉÒ¹ 2014 12:54 |
Example of group | B º .... | ¾Õª¤³Ôµ | 3 | 09 ¾ÄȨԡÒ¹ 2013 23:32 |
¾ÔÊÙ¨¹ì·ÄÉ®Õ Group | ShanaChan | ¾Õª¤³Ôµ | 1 | 16 ¸Ñ¹ÇÒ¤Á 2011 08:38 |
⨷Âìà¡ÕèÂǡѺ group | warut | ¾Õª¤³Ôµ | 10 | 21 ¸Ñ¹ÇÒ¤Á 2001 18:07 |
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