ʻO ke kumumanaʻo i kēia manawa kahi o ka makemakika. ʻIke mākou āpau i kapa ʻia ia he hoʻonohonoho ʻike maopopo ʻia kēlā me kēia ʻāpana o nā ʻē aʻe mai kekahi i kekahi, i loaʻa hoʻokahi (a ʻoi paha) mau ʻano i ka like. Hoʻonohonoho i nā kumumanaʻo i nā waiwai a me nā pilina o nā hoʻonohonoho; Ua hoʻolauna ʻia kēia kahua e Bolzano a me Cantor, a laila hoʻopau maikaʻi ʻia i ke kenekulia 20 e nā makemakika ʻē aʻe, e like me Zermelo a me Fraenkel.
He mea nui e wehewehe pono ʻia kēlā me kēia hoʻonohonoho, ʻo ia hoʻi, hiki ke hoʻokumu ʻia me ke kikoʻī inā hāʻawi ʻia i kahi mea, pili a ʻaʻole paha i ka hoʻonohonoho.
- Ma makemakika pololoi pololei kēia. ʻO kahi laʻana, inā e noʻonoʻo ʻia ka hoʻonohonoho o nā helu i ʻoi aku ma mua o ka 1 a ʻoi aku ka liʻiliʻi ma mua o 15, maopopo leʻa e hoʻopili ʻia kēia pūʻulu i nā helu 2, 4, 6, 8, 10, 12 a me 14 wale nō.
- Ma ʻōlelo maʻamau, ke kamaʻilio e pili ana i kahi hui hiki ke ʻoi aku ka pololei ʻole, no ka mea inā makemake mākou e hoʻokumu i ka hui o nā mea hīmeni ʻoi loa, no ka laʻana, ʻokoʻa nā manaʻo a ʻaʻohe ʻaelike kūlike i ka mea e lilo i ʻāpana o kēia hui a ʻo wai ka makemake ʻole. . ʻO kekahi o nā pūʻulu kūikawā he mau seta kau (ʻaʻohe o nā ʻaoʻao) a i ʻole nā ʻāpana unitary (me hoʻokahi wale nō heheʻe).
ʻO ka kapa ʻia nā mea he ʻāpana o kahi hoʻonohonoho i lālā a i ʻole ʻūmeni, a hōʻike 'ia nā seti i nā kikokikona i kākauʻia i loko o nā pale kuhi: {}. Ma loko o ka paʻa, hoʻokaʻawale ʻia nā mea e nā koma. Hiki iā lākou ke pani ʻia e nā kiʻiona Venn, kahi e hoʻopili ai i nā hōʻiliʻili o nā mea e hana i kēlā me kēia hoʻonohonoho i kahi laina paʻa a paʻa hoʻi, ma ke ʻano o kahi pōʻai. Ke loaʻa kekahi o kēia mau laina pani, hāʻawi ʻia kēlā me kēia o lākou i kahi leka nui (A, B, C, a me nā mea ʻē aʻe) a ʻo ka hoʻonohonoho honua o kēia mau mea e hōʻike ʻia e ka leka U, ʻo ia hoʻi ka hoʻonohonoho ākea.
Me nā hoʻonohonoho hiki iā ʻoe ke hana nā hana; ʻo ka mea nui ka hui ʻana, ke kuʻina, ka ʻokoʻa, ka hoʻopiha a me ka huahana Cartesian. Ua wehewehe ʻia ka hui ʻana o nā set A a me B e like me ka set A ∪ B a loaʻa kēia i kēlā me kēia mea i loko o hoʻokahi o lākou. ʻO ka hoʻohālikelike maʻamau e hōʻike ana iā ia:
- TO= {José, Jerónimo}, ʻO B= {María, Mabel, Marcela}; AUB= {José, Jerónimo, María, Mabel, Marcela}
- P= {pea, ʻāpala}, C= {lemona, ʻalani}; F= {cherry, currant};PUCUF = {pear, apple, lemon, ʻalani, cherry, currant}
- M={7, 9, 11}, N={4, 6, 8}; MANU={7, 9, 11, 4, 6, 8}
- R= {kinipōpō, skate, hoe}, ʻO G= {hoe, kinipōpō, skate}; RUG= {kinipōpō, hoe, skate}
- C= {daisy}, S= {carnation}; CUS = {daisy, hoʻokalakupua}
- C= {daisy}, S= {carnation}; T= {ʻōmole}, CUSUT = {margarita, ʻōkalakala, ʻōmole}
- ʻO G= {ʻōmaʻomaʻo, polū, ʻeleʻele}, H= {ʻeleʻele}; GUH= {ʻōmaʻomaʻo, polū, ʻeleʻele}
- TO={ 1, 3, 5, 7, 9 }; ʻO B={ 10, 11, 12 }; AUB={ 1, 3, 5, 7, 9, 10, 11, 12 }
- D= {Pōʻalua, Pōʻahā}, A ME= {Pōʻakolu, Pōʻalima}; LOA = {Pōʻalua, Pōʻakolu, Poʻahā, Pōʻalima}
- ʻO B= {mosquito, bee, hummingbird}; C= {pipi, ʻīlio, lio}; BUC= {makika, pi, hummingbird, bipi, ʻīlio, lio}
- TO={2, 4, 6, 8}, ʻO B={1, 2, 3, 4}; AUB={1, 2, 3, 4, 6, 8}
- P= {pākaukau, noho}, Q= {pākaukau, noho}; PUQ= {pākaukau, noho}
- TO= {berena}, B = {tī}; AUB= {berena, tī}
- TO={20, 30, 40}, ʻO B= {5, 15}; AUB ={5, 15, 20, 30, 40}
- M= {Ianuali, Pepeluali, Malaki, ʻApelila}, N= {Nowemapa, Kekemapa}; MANU= {Ianuali, Pepeluali, Malaki, ʻApelila, Nowemapa, Kekemapa}
- F={12, 22, 32, 42}, ʻO G= {a, e, i, o, u}; ʻO FUG= {12, 22, 32, 42, a, e, i, o, u}
- TO= {kauwela}, ʻO B= {hoʻoilo}; AUB= {kauwela, hoʻoilo}
- S= {kāmaʻa kāmaʻa, paheʻe, kāmaʻa hoʻohuli}, R= {pālule}; KA HEMA= {kāmaʻa kāmaʻa, paheʻe, kāmaʻa hoʻohuli, pālule}
- H= {Pōʻakahi, Poʻalua}, R= {Pōʻakahi, Poʻalua}, D= {Pōʻakahi, Poʻalua}; HURUDA= {Pōʻakahi, Poʻalua}
- P= {ʻulaʻula, polū}, Q= {ʻōmaʻomaʻo, melemele}, PUQ= {ʻulaʻula, uliuli, ʻōmaʻomaʻo, melemele}
