Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o nā hana Injective, Surjective, a me Bijective
ʻO ka wehewehe ʻana o kahi hana a me kona pono i ka makemakika he kumuhana hoihoi pinepine ia o ke kūkākūkā ʻana. Ma kēia ʻano, hālāwai pinepine mākou me nā huaʻōlelo e like me nā hana injective, surjective, a me bijective. He mea nui ka hoʻomaopopo ʻana i kēia mau ʻano hana ʻekolu no ka loiloi makemakika a me kā lākou hoʻohana pono ʻana ma nā ʻano like ʻole e like me ka ʻepekema kamepiula, ka hoʻokele waiwai, a me ka physics.
Ke hoʻomaopopo ʻana i nā hana Injective, Surjective, a me Bijective
Ma mua o ke kūkākūkā ʻana i nā nīnau hoʻohālike a me kā lākou kūkākūkā ʻana, e hoʻomanaʻo mua kākou i nā wehewehena o nā hana ʻekolu.
1. Hana Hoʻokomo (Hana Hoʻokahi-a-Hoʻokahi): Ua kapa ʻia kahi hana f: A → B he hoʻokomo inā no kēlā me kēia a1 a me a2 ma ke kikowaena A, inā f(a1) = f(a2), a laila pono e like ʻo a1 me a2. I nā huaʻōlelo ʻē aʻe, hōʻoia kahi hana hoʻokomo e hoʻopaʻa ʻia nā mea like ʻole ma ke kikowaena A i nā mea like ʻole ma ke kikowaena B.
2. Hana Surjective (Ma luna o ka Hana): Ua kapa ʻia kahi hana f: A → B he surjective inā loaʻa i kēlā me kēia mea i loko o ke codomain B ma ka liʻiliʻi hoʻokahi mea i loko o ke kikowaena A i hoʻopaʻa ʻia iā ia. I kēia hihia, ʻaʻohe mea "hakahaka" o ke codomain B a i ʻole nā mea like ʻole mai ke kikowaena A.
3. Hana Bijective (Hoʻopili Hoʻokahi-a-Hoʻokahi): Ua kapa ʻia kahi hana f: A → B he bijective inā he injective a he surjective hoʻi. ʻO ke ʻano kēia he hoa kū hoʻokahi ko kēlā me kēia mea ma ke kikowaena A ma ke codomain B, a he hoa kū hoʻokahi ko kēlā me kēia mea ma ke codomain B ma ke kikowaena A.
Nā nīnau hoʻohālike a me nā kūkākūkā
Nīnau 1: Hana Hoʻokomo
Nīnau:
Hāʻawi ʻia kahi hana f: ℝ → ℝ i wehewehe ʻia ʻo f(x) = 2x + 3. E hōʻoia he hana hoʻokomo kēia hana.
Kūkākūkā:
I mea e hōʻoia ai he hana hoʻokomo kēia, pono mākou e hōʻike inā ʻo f(a) = f(b) a laila ʻo a = b.
Manaʻo ʻia ʻo f(a) = f(b), ʻōlelo mākou penei:
2a + 3 = 2b + 3
E unuhi i 3 mai nā ʻaoʻao ʻelua:
\[ 2a = 2b \]
E puʻunaue i 2 ma nā ʻaoʻao ʻelua:
\[ a = b \]
No ka mea ua hōʻike mākou ʻo f(a) = f(b) ke kumu o a = b, a laila ʻo ka hana f(x) = 2x + 3 he hana hoʻokomo.
Nīnau 2: Hana Surjective
Nīnau:
Hāʻawi ʻia kahi hana g: ℝ → ℝ i wehewehe ʻia ʻo g(x) = x^3. E hōʻoia he hana surjective kēia hana.
Kūkākūkā:
No ka hōʻoia ʻana he hana surjective kēia, pono mākou e hōʻike no kēlā me kēia element y i loko o ke codomain ℝ, aia ma ka liʻiliʻi hoʻokahi element x i loko o ke domain ℝ i hiki ai iā g(x) = y.
E hoʻohana iā y ∈ ℝ. Makemake mākou e ʻimi i ka x i penei:
\[ x^3 = y \]
E lawe i ka \( x = \sqrt[3]{y} \):
g(\sqrt[3]{y}) = (\sqrt[3]{y})^3 = y \]
ʻOiai no kēlā me kēia y i loko o ke codomain ℝ hiki iā mākou ke loaʻa iā x ʻo ia hoʻi \( x = \sqrt[3]{y} \), a laila ʻo ka hana g(x) = x^3 he hana surjective.
Nīnau 3: Nā Hana Bijective
Nīnau:
Hāʻawi ʻia kahi hana h: ℝ → ℝ i wehewehe ʻia ʻo h(x) = x – 1. E hōʻoia he bijective kēia hana.
Kūkākūkā:
Hoʻokomo:
I mea e hōʻoia ai he injective ʻo h(x), pono mākou e hōʻike inā ʻo h(a) = h(b) a laila ʻo a = b.
ʻO h(a) = h(b):
\[ a – 1 = b – 1 \]
E hoʻohui i 1 ma nā ʻaoʻao ʻelua:
\[ a = b \]
ʻOiai ʻo h(a) = h(b) ke kumu o a = b, a laila he hana hoʻokomo ka hana h(x) = x – 1.
Manaʻo Kūikawā:
No ka hōʻoia ʻana he surjective ʻo h(x), pono mākou e hōʻike no kēlā me kēia element y i loko o ke codomain ℝ, aia ma ka liʻiliʻi hoʻokahi element x i loko o ke domain ℝ i hiki ai iā h(x) = y.
E hoʻohana iā y ∈ ℝ. Makemake mākou e ʻimi i ka x i penei:
\[ x – 1 = y \]
E hoʻohui i 1 ma nā ʻaoʻao ʻelua:
\[ x = y + 1 \]
No ka mea, no kēlā me kēia y i loko o ke codomain ℝ hiki iā kākou ke loaʻa kahi x i hiki ai iā x = y + 1, a laila ʻo ka hana h(x) = x - 1 he hana surjective.
No ka mea he injective a he surjective ʻo h(x), a laila he hana bijective ʻo h(x).
Nīnau 4: Ke hoʻoholo nei i ke ʻano o ka hana
Nīnau:
Hāʻawi ʻia kahi hana f: ℕ → ℕ i wehewehe ʻia ʻo f(x) = 2x. E hoʻoholo inā he hana injective, surjective, a bijective paha ʻo f.
Kūkākūkā:
Hoʻokomo:
I mea e hōʻoia ai he hana hoʻokomo kēia, pono mākou e hōʻike inā ʻo f(a) = f(b) a laila ʻo a = b.
Manaʻo ʻia ʻo f(a) = f(b):
\[ 2a = 2b \]
E puʻunaue i 2 ma nā ʻaoʻao ʻelua:
\[ a = b \]
No laila, he hana hoʻokomo ʻo f(x) = 2x.
Manaʻo Kūikawā:
No ka hōʻoia ʻana he hana surjective kēia, pono mākou e hōʻike no kēlā me kēia element y i loko o ke codomain ℕ, aia ma ka liʻiliʻi hoʻokahi element x i loko o ke domain ℕ i hiki ai iā f(x) = y.
Akā e hoʻomaopopo ʻo ke codomain ʻo ℕ (nā helu kūlohelohe), ʻoiai ʻo f(x) = 2x e loaʻa wale nā helu like. Manaʻo ʻia he helu ʻē ʻo y, ʻaʻohe x ma ℕ i hiki ai iā 2x = y.
No laila, ʻaʻole he hana surjective ʻo f(x) = 2x.
ʻOiai ʻaʻole ʻo f(x) he surjective, a laila ʻaʻole hoʻi ʻo f(x) he bijective.
Ma muli o nā hiʻohiʻona like ʻole ma luna, hiki iā mākou ke ʻike pehea e hōʻoia a ʻike ai i nā ʻano hana (injective, surjective, bijective) mai nā wehewehe hana like ʻole. He mea koʻikoʻi ka hoʻomaopopo ʻana i kēia mau hana ma nā ʻano he nui o ka makemakika a me kāna mau noi maoli.
