Nā Kumu o ke Kumumanaʻo Hoʻonohonoho
ʻO ke kumumanaʻo hoʻonohonoho kekahi o nā kumu koʻikoʻi o ka makemakika hou. Kokoke i kēlā me kēia lālā o ka makemakika—mai ka algebra a me ka loiloi a hiki i ka probability a me nā helu helu a hiki i ka ʻepekema kamepiula—hoʻohana i ke kumumanaʻo o nā set e wehewehe i nā mea, kūkulu i nā ʻano, a kūkulu i nā hoʻopaʻapaʻa logical. ʻO ka hoʻomaopopo ʻana i nā kumu o ke kumumanaʻo set e maʻalahi ai ke aʻo ʻana i nā manaʻo makemakika holomua, ʻoiai he nui nā wehewehe kūhelu e puka mai ana mai ke ʻano o kā mākou hui ʻana a me ka hoʻoponopono ʻana i nā "hōʻiliʻili" o nā mea.
1. Ke Hoʻomaopopo ʻana i nā Sets a me ko lākou mau lālā
I ka ʻōlelo maʻalahi, he hōʻiliʻili maopopo o nā mea kahi set. Ua kapa ʻia nā mea i loko o kahi set he mau lālā a he mau element paha. He mea nui ka maopopo o ka wehewehe ʻana: pono mākou e hiki ke hoʻoholo inā he lālā o ka set kahi mea a ʻaʻole paha.
laʻana:
– ʻO ka hoʻonohonoho o nā helu like ma lalo o 10 he {2, 4, 6, 8}.
– ʻO ka hoʻonohonoho ʻana o nā leo kani ma ka ʻōlelo Indonesia ʻo {a, i, u, e, o}.
Nā hōʻailona i hoʻohana pinepine ʻia:
– Inā he lālā ʻo \(x\) o ka set \(A\), e kākau iā \(x \in A\).
– Inā ʻaʻole ʻo \(x\) he lālā o \(A\), ua kākau ʻia ʻo \(x \notin A\).
Eia kekahi laʻana, inā \(A = \{1,2,3\}\), a laila \(2 \in A\) a me \(5 \notin A\).
2. Pehea e hōʻike ai i kahi Set
Aia kekahi mau ala e hōʻike ai i kahi set:
1. Ma ke kākau inoa ʻana i nā lālā (ʻano papa inoa)
Laʻana: \(A = \{1,2,3,4\}\).
2. Me ka wehewehe (hōʻailona hoʻonohonoho-mea kūkulu)
Laʻana: \(B = \{x \mid x \text{ helu kūlohelohe a me } x < 5\}\). Heluhelu ʻia penei: "ʻO B ka hoʻonohonoho o nā \(x\) āpau i hiki ai iā \(x\) ke helu kūlohelohe a me \(x < 5\)."
3. Me nā kiʻikuhi Venn, hōʻike nā kiʻikuhi Venn i nā pilina ma waena o nā seti me ka hoʻohana ʻana i nā kinona (ʻo nā pōʻai maʻamau) i loko o ke ao holoʻokoʻa o ke kūkākūkā ʻana. Aia ke koho ʻana o ke ʻano hōʻike i nā pono: kūpono ka papa inoa no nā seti liʻiliʻi, ʻoiai kūpono ka hōʻailona kūkulu seti no nā seti nui a palena ʻole paha. 3. Set Universal a me ka Set Empty Ma kekahi mau kūkākūkā ʻana, wehewehe pinepine mākou i ka set universal \(U\), ʻo ia ka set e loaʻa ana nā mea āpau e kūkākūkā ʻia nei. No ka laʻana, inā mākou e kūkākūkā ana i nā helu helu, a laila hiki i ke ao holoʻokoʻa ke \(U = \mathbb{Z}\). I kēia manawa, ʻo ka set hakahaka he set ʻaʻohe ona lālā, i hōʻailona ʻia e \(\varnothing\) a i ʻole \(\{\}\). He laʻana o kahi set hakahaka: ʻo ka set o nā helu kūlohelohe ma lalo o 0. ʻAʻohe helu kūlohelohe e hoʻokō i kēlā kūlana, no laila ua hakahaka ka set. 4. Ka like o nā Sets Ua ʻōlelo ʻia ʻelua mau set he like inā loaʻa iā lākou nā lālā like. ʻAʻole koʻikoʻi ke kaʻina o ke kākau ʻana i nā lālā. Laʻana: - \(\{1,3,5\} = \{5,3,1\}\) ʻAʻole like me nā papa inoa maʻamau, ʻaʻole mālama nā set i ke kauoha a ʻaʻole helu i nā kope. No laila: - \(\{1,1,2,2,3\} = \{1,2,3\}\) 5. Nā Subsets a me nā Subsets Kūpono Inā he mau mea hoʻi nā mea āpau o kahi set \(A\) o kahi set \(B\), a laila ua kapa ʻia ʻo \(A\) he subset o \(B\), i kākau ʻia ʻo \(A \subseteq B\). Laʻana: - Inā ʻo \(B = \{1,2,3,4\}\) a me \(A = \{2,4\}\), a laila ʻo \(A \subseteq B\). Inā he subset o \(B\) ʻo \(A\) akā ʻaʻole like ʻo \(A\) me \(B\), a laila ua kapa ʻia ʻo \(A\) he subset maoli, i kākau ʻia ʻo \(A \subset B\).
ʻOiaʻiʻo koʻikoʻi: ʻO ka set hakahaka he ʻāpana liʻiliʻi o kēlā me kēia set, ʻo ia hoʻi, \(\varnothing \subseteq A\) no kekahi set \(A\). 6. Nā Hana Kumu ma nā Sets Hāʻawi ke kumumanaʻo set i nā hana no ka hoʻohui ʻana a i ʻole ka hoʻohālikelike ʻana i nā set. a) Union ʻO ka union \(A \cup B\) ka set e loaʻa ana nā mea āpau i loko o \(A\) a i ʻole i loko o \(B\) (a i ʻole i nā mea ʻelua). Laʻana: - \(A = \{1,2,3\}\), \(B = \{3,4,5\}\) A laila \(A \cup B = \{1,2,3,4,5\}\). b) Intersection Aia ka intersection \(A \cap B\) i nā mea i loko o \(A\) a me \(B\). Laʻana: - \(A \cap B = \{3\}\). c) ʻOkoʻa Loaʻa ka ʻokoʻa \(A - B\) (a i ʻole \(A \setminus B\)) i nā mea i loko o \(A\) akā ʻaʻole i loko o \(B\). Laʻana: - \(A \setminus B = \{1,2\}\). d) Hoʻopihapiha ʻO ka hoʻopihapiha o \(A^c\) (a i ʻole \(\overline{A}\)) ʻo ia ka element o ke ao holoʻokoʻa \(U\) ʻaʻole i hoʻokomo ʻia i loko o \(A\). Laʻana: inā \(U = \{1,2,3,4,5\}\) a me \(A = \{1,3\}\), a laila \(A^c = \{2,4,5\}\). 7. Nā Kānāwai Koʻikoʻi i nā Hana Hoʻonohonoho Loaʻa i nā hana hoʻonohonoho nā waiwai like me nā hana ma nā helu. 1. Commutative \(A \cup B = B \cup A\) a me \(A \cap B = B \cap A\). 2. Associative \((A \cup B) \cup C = A \cup (B \cup C)\) \((A \cap B) \cap C = A \cap (B \cap C)\). 3. Hoʻolaha \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\).
4. Nā Kānāwai o De Morgan \((A \cup B)^c = A^c \cap B^c\) \((A \cap B)^c = A^c \cup B^c\). He mea pono loa kēia mau kānāwai i ka hoʻomaʻalahi ʻana i nā hōʻike set, ʻoiai ke hana nei me ka logic, probability, a me nā ʻano algebraic. 8. Cardinality: Helu o nā ʻElemene o kahi Set ʻO Cardinality ka helu o nā ʻelemene i loko o kahi set, i hōʻike ʻia e \(|A|\). No nā set palena, maʻalahi ka helu ʻana i ke cardinality. Laʻana: - Inā \(A = \{2,4,6\}\), a laila \(|A| = 3\). No nā set palena ʻole, lilo ka manaʻo o ke cardinality i mea hoihoi (no ka laʻana, ʻo ka set o nā helu kūlohelohe \(\mathbb{N}\) he cardinality palena ʻole). Eia naʻe, hele pinepine kāna kūkākūkā ʻana i ke kumumanaʻo set holomua. 9. Huahana Cartesian a me nā Pilina Maʻalahi ʻO ka huahana Cartesian o \(A\) a me \(B\), i kākau ʻia ʻo \(A \times B\), ʻo ia ka hui o nā hui i kauoha ʻia \((a,b)\) me \(a \in A\) a me \(b \in B\). Laʻana: - Inā \(A = \{1,2\}\) a me \(B = \{x,y\}\), a laila \(A \times B = \{(1,x),(1,y),(2,x),(2,y)\}\). ʻO ka huahana Cartesian ke kumu no ke aʻo ʻana i nā pilina a me nā hana, no ka mea, hiki ke nānā ʻia nā hana ma ke ʻano he mau hui o nā hui i kauoha ʻia me kekahi mau lula. Hopena ʻO nā kumu o ke kumumanaʻo set e aʻo mai iā mākou pehea e hoʻonohonoho ai i nā mea ma ke ʻano i hoʻonohonoho ʻia a kūlike. Ma ka hoʻomaopopo ʻana i nā manaʻo o nā mea, nā subsets, nā hana union/intersection/difference/complement, nā kānāwai o nā hana, a me nā manaʻo o ke cardinality a me ka huahana Cartesian, loaʻa iā mākou nā mea hana koʻikoʻi e neʻe ai i nā kumuhana makemakika holomua. ʻAʻole wale ke kumumanaʻo hoʻonohonoho he mea kumu, akā he ʻōlelo honua hoʻi i hoʻohana ʻia ma nā ʻano ʻepekema a me ka ʻenehana he nui. ʻO ka hoʻokele pono ʻana i kēia mau manaʻo e maʻalahi a ʻoi aku ka loina o ke aʻo ʻana i ka makemakika ma hope.
