Nā Hana Injective, Surjective, a me Bijective
Ma ka makemakika, ʻoi aku hoʻi ma ke kumumanaʻo hana, aia ʻekolu ʻano hana koʻikoʻi e kūkākūkā pinepine ʻia: injective, surjective, a me bijective. Loaʻa i kēlā me kēia o kēia mau ʻano hana ʻekolu nā ʻano kūikawā e hoʻoholo ai pehea e hoʻohālikelike ʻia ai nā mea mai ka set kumu (domain) i nā mea i loko o ka set target (range a i ʻole codomain). E wehewehe kēia ʻatikala i ka wehewehe ʻana, nā waiwai, a me nā laʻana o kēlā me kēia o kēia mau hana, a me kā lākou noi ma nā ʻano like ʻole.
Hana Hoʻokomo
ʻO kahi hana hoʻokomo, i ʻike ʻia hoʻi he hana hoʻokahi-a-hoʻokahi, he hana ia e hoʻohālikelike ʻia ai kēlā me kēia mea i loko o ka hoʻonohonoho kumu i kahi mea kū hoʻokahi i loko o ka hoʻonohonoho hopena. Ma ke ʻano maʻamau, ua kapa ʻia kahi hana \( f : A \to B \) he hoʻokomo inā a inā wale nō no kēlā me kēia \( a_1, a_2 \in A \), \( f(a_1) = f(a_2) \) e hōʻike ana ʻo \( a_1 = a_2 \).
ʻOi aku ka maʻalahi, hōʻoia kahi hana injective ʻaʻohe ʻelua mau mea like ʻole i loko o ka hoʻonohonoho kumu i loaʻa ke kiʻi like ma ka hoʻonohonoho hopena. I nā huaʻōlelo ʻē aʻe, ʻo kēlā me kēia mea i loko o ka hoʻonohonoho hopena he hoʻokahi wale nō mea kumu e pili ana iā ia.
laʻana:
– E noʻonoʻo i ka hana \( f: \mathbb{R} \to \mathbb{R} \) i wehewehe ʻia ʻo \( f(x) = 2x + 3 \). He hana hoʻokomo kēia no ka mea inā \( f(a) = f(b) \), a laila \( 2a + 3 = 2b + 3 \), ʻo ia hoʻi \( a = b \).
noi:
Hoʻohana pinepine ʻia nā hana injective i nā pōʻaiapili kahi e pono ai mākou e hōʻoia ʻaʻohe hana pālua, e like me ka indexing a i ʻole ke coding.
Hana Surjective
ʻO ka hana surjective, a i ʻole onto-function, he hana ia e loaʻa ai i kēlā me kēia mea i loko o ka set destination \( B \) ma ka liʻiliʻi hoʻokahi mea mai ka set source \( A \) e pili ana iā ia. Ma ka notation formal, ua kapa ʻia kahi hana \( f : A \to B \) he surjective inā no kēlā me kēia \( b \in B \), aia ma ka liʻiliʻi hoʻokahi \( a \in A \) i hiki ai \( f(a) = b \).
ʻO ia hoʻi, hōʻoia ka hana surjective ua uhi piha ʻia ka set destination e ke kiʻi o ka set source. ʻAʻohe mea i loko o ka set destination i "uhi ʻia."
laʻana:
– E noʻonoʻo i ka hana \( f: \mathbb{R} \to \mathbb{R} \) i wehewehe ʻia ʻo \( f(x) = x^3 \). He hana surjective kēia no ka mea no kēlā me kēia \( y \in \mathbb{R} \), hiki iā mākou ke loaʻa \( x \in \mathbb{R} \) i hiki ai \( x^3 = y \).
noi:
Hoʻohana nui ʻia nā hana surjective i loko o ka pōʻaiapili o ka hoʻokaʻawale ʻana a i ʻole ka hoʻokaʻawale ʻana o nā kumuwaiwai, kahi e pono ai mākou e hōʻoia e loaʻa ana i kēlā me kēia mea loaʻa kekahi mea mai ka hui o nā mea hāʻawi.
Hana Bijective
ʻO ka hana bijective he hana injective a surjective hoʻi. I nā huaʻōlelo ʻē aʻe, ʻo ka hana bijective he hoʻokahi-i-hoʻokahi a me ke onto. No laila, i loko o kahi hana bijective, ua hoʻohālikelike ʻia kēlā me kēia mea i loko o ka set kumu i kahi mea i loko o ka set hopena, a i ka ʻaoʻao ʻē aʻe, loaʻa i kēlā me kēia mea i loko o ka set hopena hoʻokahi wale nō mea e hoʻohālikelike ana iā ia mai ka set kumu.
laʻana:
– E noʻonoʻo i ka hana \( f: \mathbb{R} \to \mathbb{R} \) i wehewehe ʻia e like me \( f(x) = x + 1 \). He bijective kēia hana no ka mea:
– Hoʻokomo: Inā \( f(a) = f(b) \), a laila \( a + 1 = b + 1 \), hōʻike ia \( a = b \).
– Surjective: No kēlā me kēia \( y \in \mathbb{R} \), hiki iā mākou ke loaʻa \( x = y – 1 \) i hiki ai iā \( f(x) = y \).
noi:
He mea nui loa nā hana bijective i loko o ka pōʻaiapili o nā hoʻololi a me nā isomorphisms, kahi e pono ai mākou e mālama i ke ʻano a i ʻole nā pilina ma waena o nā mea i ka hoʻohālikelike ʻana mai kekahi set a i kekahi. No ka laʻana, i loko o ka cryptography, ʻo nā kī encryption a me decryption he mau hana bijective pinepine i hiki ke hoʻopāʻālua ʻia a wehe ʻia nā leka.
Ka Nānā Hou Aʻe
Nā Kiʻi a me nā Kiʻikuhi
ʻO ka hoʻohana ʻana i kahi kiʻikuhi Venn a i ʻole ka pakuhi he mea kōkua pinepine ia e hoʻomaopopo i kēia mau hana. Ma kahi kiʻikuhi Venn, hiki ke hōʻike ʻia kahi hana injective e kēlā me kēia mea i loko o ka set destination me ka nui o hoʻokahi pua e hiki mai ana. Hiki ke hōʻike ʻia kahi hana surjective e kēlā me kēia mea i loko o ka set destination me ka liʻiliʻi o hoʻokahi pua e hiki mai ana. Loaʻa i kahi hana bijective kēlā me kēia mea i loko o nā set source a me destination i hoʻokahi pua e hiki mai ana, e hana ana i kahi pilina hoʻokahi-a-hoʻokahi.
Hana Hoʻohuli
ʻO kekahi ʻano koʻikoʻi i aʻo pinepine ʻia ma ke ʻano o nā hana injective, surjective, a me bijective ʻo ia ka hana inverse.
– Loaʻa mau i kahi hana injective kahi hana inverse hema.
– Loaʻa mau i kahi hana surjective kahi hana inverse pololei.
– Loaʻa mau i kahi hana bijective kahi hana inverse kū hoʻokahi.
Inā he bijective kahi hana, e loaʻa nā inverses hema a me ka ʻākau a e like nā mea ʻelua, e hana ana i ka hana inverse maoli.
Pani
ʻO ka hoʻomaopopo ʻana i nā manaʻo o nā hana injective, surjective, a me bijective he mea nui ia i nā lālā he nui o ka makemakika a me kā lākou mau noi kūpono. Hōʻoia nā hana injective ʻaʻole e hana pālua; hōʻoia nā hana surjective i ka uhi piha ʻana; a hōʻoia nā hana bijective i ka pilina hoʻokahi-a-hoʻokahi ma waena o nā mea i loko o ʻelua mau seti. He mea nui ka ʻike o kēia mau ʻano hana ʻekolu ʻaʻole wale ma ka makemakika maʻemaʻe akā ma nā kahua e like me ka ʻepekema kamepiula, ka hoʻokele waiwai, a me ka ʻenekinia. ʻO ka hoʻomaopopo piha ʻana i ka hana a me nā noi o kēia mau hana hiki ke wehe i ka puka i ka loiloi a me ka hoʻoponopono pilikia ʻoi aku ka maikaʻi a me ka maikaʻi.
