Nā ʻAno Hōʻoia Makemakika
ʻO ka hōʻoia ma ka makemakika ke kikowaena o kēia aʻo ʻana. ʻO nā ʻano hōʻoia ke kumu no ka hōʻoia ʻana i ka ʻoiaʻiʻo o kahi ʻōlelo makemakika. Mai nā kuhiakau kumu a hiki i nā hopena, pono e hōʻoia ʻia kēlā me kēia ʻanuʻu he kūpono. ʻO ka hoʻomaopopo ʻana i nā ʻano hōʻoia like ʻole ʻaʻole wale e hoʻoikaika i nā mākau loiloi akā hoʻonui pū i ka ʻike aʻo a me ka hoʻopili ʻana o ka makemakika ma nā ʻano like ʻole.
E kūkākūkā kēia ʻatikala i kekahi o nā ʻano hōʻoia nui i ka makemakika, me ka hōʻoia pololei, ka hōʻoia ʻaʻole pololei (contraposition a me contraction), ka hoʻokomo makemakika, a me ka hōʻoia ma o ka laʻana kikoʻī. Loaʻa i kēlā me kēia ʻano hana nā noi like ʻole, nā ikaika, a me nā nāwaliwali. E ʻimi hohonu kākou iā lākou.
1. Hōʻike Pololei
Wehewehena a me nā Laʻana
ʻO ka hōʻoia pololei kahi ʻano hana e hōʻoia ai mākou i kahi ʻōlelo ma ka hōʻike ʻana inā he ʻoiaʻiʻo nā kumumanaʻo (nā kuhiakau), a laila he ʻoiaʻiʻo nō hoʻi ka hopena. I ka hōʻoia pololei, hoʻomaka pinepine mākou me ka mea i ʻike ʻia a hoʻohana i nā ʻanuʻu kūpono e hiki ai i ka hopena.
laʻana:
E hōʻoia inā he helu like ʻo \(n\), a laila he helu like nō hoʻi ʻo \(n^2\).
Hōʻike:
Manaʻo ʻia he helu like ʻo \(n\). A laila, e like me ka wehewehe ʻana o kahi helu like, hiki ke kākau ʻia ʻo \(n = 2k\) no kekahi helu holoʻokoʻa \(k\). No laila,
\[ n^2 = (2k)^2 = 4k^2 = 2(2k^2) \]
Ua maopopo he hiki ke hōʻike ʻia ʻo \(n^2\) ma ke ʻano he 2 manawa o ka helu holoʻokoʻa (ʻo ia hoʻi \(2k^2\)). ʻOiai ʻo ke koi nui no kahi helu like, ʻo ia ka hiki ke hōʻike ʻia he 2 manawa o ka helu holoʻokoʻa, a laila he helu like nō hoʻi ʻo \(n^2\).
2. Nā Hōʻike Kūlike ʻole
ʻO ka hōʻoia pololei ʻole e pili ana i ʻelua mau ʻano nui: ka hōʻoia ma o ka kūʻē ʻana a me ka hōʻoia ma o ka kūʻē ʻana.
a. Hōʻike o ka Contraposition
Wehewehena a me nā Laʻana
ʻO kēia ʻano hana e pili ana i ka hōʻoia ʻana i ka ʻōlelo hoʻopili "inā \(P\), a laila \(Q\)" ma ka hōʻoia ʻana i ka contrapositive o ka ʻōlelo: "inā ʻaʻole \(Q\), a laila ʻaʻole \(P\)".
laʻana:
E hōʻoia inā he ʻē ka \(n^2\), a laila he ʻē nō hoʻi ʻo \(n\).
Hōʻike:
ʻO ke kū'ē o ka ʻōlelo penei: Inā ʻaʻole ʻē (a ʻaʻole like), a laila ʻaʻole ʻē (a ʻaʻole like) ʻo \(n^2\).
Manaʻo ʻia he like ʻo \(n\), a laila ʻo \(n = 2k\) no kahi helu holoʻokoʻa \(k\). No laila,
\[ n^2 = (2k)^2 = 4k^2 = 2(2k^2) \]
ʻO ke ʻano kēia he helu like ʻo \(n^2\). No laila, ua hōʻoia ʻia ka contrapositive, a ua hōʻoia ʻia hoʻi ka ʻōlelo mua he ʻoiaʻiʻo.
b. Hōʻoia ma o ke Kūʻē
Wehewehena a me nā Laʻana
ʻO ka hōʻoia ma ke kūʻē ʻana e pili ana i ka manaʻo ʻana he wahaheʻe ka ʻōlelo e hōʻoia ʻia a me ka hōʻike ʻana e alakaʻi ana kēia kuhi i kahi kūʻē kūpono.
laʻana:
E hōʻoia i ka helu irrational ʻo \(\sqrt{2}\).
Hōʻike:
E kuhi, ma kahi o ia mea, he helu kūpono ʻo \(\sqrt{2}\). A laila, ʻo \(\sqrt{2} = \frac{a}{b}\), kahi ʻo \(a\) a me \(b\) he mau helu piha (ʻo ka unuhi ʻana he 1), a ʻo \(b \ne 0\). No laila, hiki iā mākou ke kākau:
\[ \sqrt{2} = \frac{a}{b} \]
\[ 2 = \frac{a^2}{b^2} \]
\[ 2b^2 = a^2 \]
Mai kēia kaulike, ʻike mākou he helu like ʻo \(a^2\), ʻo ia hoʻi, pono e like pū ʻo \(a\). Manaʻo ʻia ʻo \(a = 2k\), loaʻa iā mākou:
2b^2 = (2k)^2
2b^2 = 4k^2
\[ b^2 = 2k^2 \]
ʻOiai he helu like ʻo \(b^2\), a laila pono nō hoʻi ʻo \(b\) he helu like. ʻO ke ʻano kēia he mau helu like ʻo \(a\) a me \(b\), e kūʻē ana i ka manaʻo mua ʻo \(\frac{a}{b}\) ma kona ʻano maʻalahi loa. No laila, ʻaʻole hiki iā \(\sqrt{2}\) ke lilo i helu kūpono, a no laila he mea kūpono ʻole ia.
3. Hoʻokomo Makemakika
Wehewehena a me nā Laʻana
ʻO ka hoʻokomo makemakika kahi ʻano hōʻoia i hoʻohana ʻia e hōʻoia i nā ʻōlelo e pili ana i nā helu helu. Aia ke kaʻina hana i ʻelua mau ʻanuʻu: ke kumu hoʻokomo a me ke ʻanuʻu hoʻokomo.
laʻana:
E hōʻoia i ka huina o ka moʻo mua o nā helu piha \(1 + 2 + 3 + … + n = \frac{n(n+1)}{2}\).
Hōʻike:
– Kumu Hoʻokomo:
No \(n = 1\),
\[ 1 = \frac{1(1+1)}{2} \]
pololei.
– Nā Kaʻina Hoʻokomo:
Manaʻo ʻia he ʻoiaʻiʻo ka ʻōlelo no kahi helu \(k\). ʻO ia hoʻi,
\[ 1 + 2 + 3 + … + k = \frac{k(k+1)}{2} \]
Pono mākou e hōʻoia he ʻoiaʻiʻo nō hoʻi ia no \(k + 1\). Hoʻohui mākou iā \((k + 1)\) i nā ʻaoʻao ʻelua o ka hoohalike:
\[ 1 + 2 + 3 + … + k + (k + 1) = \frac{k(k+1)}{2} + (k + 1) \]
\[ = \frac{k(k+1) + 2(k+1)}{2} \]
\[ = \frac{(k + 1)(k + 2)}{2} \]
No laila, he ʻoiaʻiʻo ka ʻōlelo no \(k + 1\). No laila, ma ke kumumanaʻo o ka hoʻokomo makemakika, he ʻoiaʻiʻo ka ʻōlelo no nā helu helu maikaʻi āpau \(n\).
4. Hōʻoia me nā Laʻana Kikoʻī
Wehewehena a me nā Laʻana
ʻO kēia ʻano hana e pili ana i ka hōʻoia ʻana ma ke koho ʻana i nā laʻana kikoʻī e hoʻokō ana i nā kūlana āpau i hāʻawi ʻia ma ka ʻōlelo a hōʻike i ka ʻoiaʻiʻo o ka ʻōlelo. Eia nō naʻe, hoʻohana pinepine ʻia kēia ʻano hana e hōʻoia i kahi ʻōlelo wahaheʻe.
laʻana:
E hōʻoia i ka loaʻa ʻana o nā helu i hiki ʻole ke hōʻike ʻia ma ke ʻano he huina o ʻelua mau huinaha kūpono.
Hōʻike:
E hoʻāʻo e hoʻohana i ka laʻana \(3\):
Manaʻo ʻia hiki ke hōʻike ʻia ʻo \(3\) ma ke ʻano he huina o ʻelua mau huinaha kūpono, ʻo ia hoʻi ʻo \(a^2 + b^2 = 3\). Ma hope o ka hoʻāʻo ʻana i nā hui pū ʻana o nā helu āpau \(a\) a me \(b\),
1. \(a = 0\), \(b^2 = 3\) (hiki ʻole).
2. \(a = 1\), \(b^2 = 2\) (hiki ʻole).
3. \(a = 2\), \(b^2 = -1\) (hiki ʻole).
4. ʻAʻole hiki nō hoʻi nā helu maikaʻi ʻole a i ʻole nā helu ʻoi aku ma mua o 2.
Hōʻike kēia ʻaʻole hiki ke hōʻike ʻia ʻo \(3\) ma ke ʻano he huina o ʻelua mau helu huinaha. No laila, aia nā helu i hiki ʻole ke hōʻike ʻia ma ke ʻano he huina o ʻelua mau helu huinaha kūpono.
Ka hopena
Pono nā hōʻoia ma ka makemakika i nā ʻano hana like ʻole a me nā ʻanuʻu ʻōnaehana ma muli o ke ʻano o ka ʻōlelo e hōʻoia ʻia nei. ʻO ka hōʻoia pololei, ka hōʻoia ʻaʻole pololei (contrapositive a me ka contraction), ka hoʻokomo makemakika, a me nā laʻana kūikawā kekahi o nā ʻano hōʻoia mua i hoʻohana ʻia i nā kūlana like ʻole. ʻO ka hoʻomaopopo ʻana i kēia mau ʻano hana e hoʻoikaika i nā kumu o ka makemakika a kōkua iā ʻoe e ʻimi hohonu i nā lālā like ʻole o ka makemakika.
Me ka hoʻomaʻamaʻa a me ka hoʻomaopopo hohonu, e lilo nā ʻano hōʻoia makemakika i mea hana e mākaukau mau ai e hoʻohana ʻia i ka hoʻoponopono ʻana i nā pilikia makemakika paʻakikī.
