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- Aã¨Bã®ç´ç©ã¯ãAÃB
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éåã¨ã¯ãã£ã¦ããããã§ã¯æééåã ããä¾ã«ä½¿ããã¨ã«ãã¾ããããã¦ã[0] = {}, [1] = {1}, [2] = {1, 2}, [3] = {1, 2, 3} ãªã©ã¨ç´æãã¾ããã¤ã¾ãã[n]ã¨æ¸ãã¦ãã£ããããã¯ã1ããnã¾ã§ã®æ´æ°ã®éã¾ããã§ãããããã£ã[n]ã¯æééåã®å ¸åä¾ã¨ãããã§ãããã
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å½ç¶ãªããã#[0] = 0, #[1] = 1, #[2] = 2, #[3] = 3ãä¸è¬ã« #[n] = n ã§ãããããã説æãããã¨ã¯ã
- #([n]ï¼[m]) = #[n]ï¼#[n] = nï¼m
- #([n]Ã[m]) = #[n]Ã#[n] = nÃm
- #([m][n]) = #[m]#[n] = mn
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ç´ç©ãä¸çªç°¡åãããªã®ã§ãããããã¯ããã¾ããããAÃBã¯ãAã®è¦ç´ aã¨Bã®è¦ç´ bã®çµ(a, b)ãå ¨é¨å¯ãéããéåã§ããä¾ãã°ã
- [2]Ã[3] = {1, 2}Ã{1, 2, 3} = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}
ã§ãã(1, ?) 㨠(2, ?) ã¨ãã2ã¤ã®ãã¿ã¼ã³ãèãã¦ãã¯ã¨ã¹ãã§ã³ãã¼ã¯ã®ã¨ããã«ã1, 2, 3ãå ¥ãã¨ãçµã¿åããæ¹ã¯ 2Ã3 ã§ããããããã¯ã2æ¬ã®ç¸¦ç·ã¨3æ¬ã®æ¨ªç·ã交差ãããç¢ç¤ç®ãåå®ã㦠2Ã3 ããã®äºæ ãä¸è¬åããã°ã
- #([n]Ã[m]) = #[n]Ã#[m] = nÃm
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ç´å Aï¼B ã¯ãåä½µ AâªB ã¨ã¯éãã¾ããAã¨Bã«å ±éã«å±ããè¦ç´ ããã£ã¦ããããããå¥ã ãªãã®ã ã¨æã£ã¦å¯ãéãã¾ããä¾ãã°ã[2]ï¼[3] ã¯ãåä½µ [2]âª[3] ã¨ã¯éãã¾ãã[2]âª[3] = {1, 2}âª{1, 2, 3} = {1, 2, 3} ã§ããã[2]ï¼[3] = {1, 2, 1', 2', 3} ã§ãã1' ã¯ã1 ã¨åãã¢ãã§ããå¥ç©æ±ããã¾ã -- ãªããæ¥æ¬èªããããã§ããããã§ãã¾ã¼ããããªæããªãã§ããã2' 㯠2 ã¨åãã¢ãã ãã©å¥ç©ã :-)
ãªãã ãç¦ åçã¿ããã ã¨æã人ã®ããã«ãAï¼B = ({1}ÃA)âª({2}ÃB) ã¨ããå®ç¾©ãããã¾ãããã®å®ç¾©ããããããããã¨ã¯å ¨ç¶ãªãï¼ä»ã«ãå®ç¾©ã®ä»æ¹ã¯ããï¼ã®ã§ããã便å©ãªã®ã¯ç¢ºãããã®å®ç¾©ã«ããã°ã
- {1, 2}ï¼{1, 2, 3} = ({1}Ã{1, 2})âª({2}Ã{1, 2, 3})
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- {(1, 1), (1, 2)}âª{(2, 1), (2, 2), (2, 3)} = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3)}
ã§ããããã§ã(1, ?) (2, ?) ã«åºã¦ãã1ã¨2ã¯ãç´ç©ã®å ´åã¨ã¯å ¨ç¶å¥ã§ãåºèº«ã示ãç®å°ã§ããAï¼Bããã®æ¹æ³ã§ä½ã£ãã¨ãã(1, x) ã¯Aããæ¥ãï¼Aã«ç±æ¥ããï¼xã示ãã(2, x) ã¯Bããæ¥ãxã示ãã¾ãããåãã¢ãã§ããå¥ç©æ±ããã¾ããã¨ããå¤ãªæ¥æ¬èªã¯ããåãxã§ãããç®å°ã¨ãªã1ã¨2ã§åºå¥ãã¦æ±ãã¾ããã£ã¦ãã¨ãã
{1}ÃA ã¯Aã®ã³ãã¼ã{2}ÃBã¯Bã®ã³ãã¼ãç®å°1, 2ãéãããå ±éé¨åã¯ãªãã#(Aï¼B) = #({1}ÃA)ï¼#((2}ÃB) = #Aï¼#B ã¯ç´å¾ããã§ãããçµå±ã
- #([n]ï¼[m]) = #[n]ï¼#[m] = nï¼m
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ãã¦ã[3][2]ãèãã¾ããããããã¯ï¼ä»ã®ç¢å°ã®ç¨æ³ã¨åºå¥ã§ãããªãï¼ã[2]â[3] ã¨æ¸ãã¦ãããã®ã§ããéå{1, 2}ä¸ã§å®ç¾©ãããå¤ã{1, 2, 3}ã«åãé¢æ°fã¯ãf(1)ã¨f(2)ã§æ±ºã¾ãã¾ããf(1)ã®é¸ã³æ¹ã¯3éããf(2)ã®é¸ã³æ¹ã¯3éãã3éãã¨3éããèªç±ã«çµã¿åããã¦ããã®ã§ãåè¨ 3Ã3ï¼32éãã®å¯è½æ§ãããã¾ãããããã®çµã¿åããããããã1ã¤ã®é¢æ°ã決ããã§ã®ãé¢æ°ã®åæ°ã32ã§ãã
9åå ¨é¨ã¯ã¨ãããããï¼ã¤ããã並ã¹ã¦ããã¾ãããã
- f(1) = 1, f(2) = 1
- f(1) = 1, f(2) = 2
- f(1) = 1, f(2) = 3
- f(1) = 2, f(2) = 1
[2][3] ã ã£ãããf(1), f(2), f(3) ã®å¤ããããã2éããã¤ãªã®ã§ã2Ã2Ã2 = 23 ã§ããä¸è¬ã«ã
- #([m][n]) = #[m]#[n] = mn
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