Aeaie hoahanau a me kā lākou mau waiwai

A laulā huahelu o ka pili ma no Laʻana e puhi ia pu ma ka helu nui o ka manaÿo, mai ko lakou mau wehewehe a me ka mākau Ka Ikepili o ka pau paradox. Aʻano o ka manaÿo kūkākūkā i loko o ka 'atikala ma ka ahaaina i ka wa pau ole. 'Oiai, i ka wa e kamailio ana e pili ana i ka pālua type, ma keia mea i manaoia he aeaie pilina ma waena o kekahi mau aiaiiuo. A kekahi ma waena o nä mea a makou e olelo ai.

E like me ka rula, e hōʻike i ka aeaie ana ma R, i mea, ina xRx no kekahi waiwai o ka m ma ka mahinaʻai o R, ua kapaia reflexive ia i ka waiwai, kahi m, a m - ua i mea o ka manao, a me R mea he hoailona ia no kekahi palapala o ka pili ma waena o na kanaka . I ka Ia manawa, ina o ka hookahua a xRy® yRx, ia olelo e pili ana i symmetry moku'āina kahi ® - ka implication hoailona, i 'ano like me ka hui ana o ka "ina ... laila ..." A eia ka mea hope, deciphering palapalaia maluna penei (xRy Ùy Rz). ®xRz hōʻike e pili ana i transitive pilina, a me ka hoailona o ka u make - keia mea he ka huipūʻana.

A aeaie pili i ka mea a elua reflexive, symmetric, a me ka transitive ua kapaʻia he equivalence pili. Ka lākiō o ka [illegible] - ka hana, a me ka au [illegible] a me ka au [illegible] hoʻohuʻu ka like y me = Z. Simple aeaie kuleana pili i hiki ke eaaei pili ana i ka poe noonoo ole manaʻo hoʻopiʻi kū'ē mau hoonohonoho i loko o kekahi mea, a wale i keia hihia, e 'oi aku ke kumukuai ia ia, kuhikuhi kēia mau, ka' aoʻao, ia i loko o ka ua hihia.

Ka mea, e olelo aku i ka [illegible] palapala m i ka y me, Inā. F mea he hana o ka ciia ho'ākāka 'āpana cia xa me y. Naʻe, ka wā extrapolates. F m ma ka y me, a me y me aʻu Z, laila, i kēia hiki aku ai i ka mea i [illegible] E hoike mai ma ka m Z. A ano laʻana: ina f (x) = 2x mea i pololei ia mau 'ano ākeʻakeʻa kumu helu m, laila, ke olelo aku nei au i ka [illegible] ai maps i pūlima' ia paha o nā mea a pau integers ike i na mea he nui o na ia a pau, akā, i kēia manawa, a hiki helu. I haiia ae maluna,, ka aeaie pilina i ka poʻe reflexive, symmetric, a me ka transitive, o ka pilina o ka equivalence.

Ma muli o ka mea ma luna, i ka pilina o ka equivalence nānā 'ana i ka waiwai o aeaie pili:

• reflexivity - ka lākiō (M ~ N);
• symmetry - ina e hoʻokolokolo pono M ~ N, ma laila e e N ~ M;
• transitivity - ina elua e hoʻokolokolo pono a me M ~ N N ~ P, o ka hopena M ~ P.

E noʻonoʻo i ka pili waiwai o aeaie pili i hou au mamuli. Reflexivity - o kekahi o nā 'ano o kekahi loulou, ma kēlā me kēia hehee ai o ka hoao ana e puhi ia mea i loko o kēia like wale. No ka laʻana, ma waena o nā helu i = pela aku, a a³ me - reflexive kamaʻilio, no ka mea, he mea mau i = e = pela aku, a me ka a³, s³ me. I ka Ia manawa, i ka lākiō o ka like ole o ka> e - antireflexive no ka mea, o ka mea impossibility o ka like ole he> he. Ke axiom o keia waiwai ua encoded huapalapala: aRc® Ara U cRc, 'aneʻi ka hōʻailona ® Hōʻike i ka hua'ōlelo "hoʻohuʻu" (' ole "hoʻohuʻu") a me ka U hoailona - kū ma "a me ka" (a me ka huipūʻana). Mai keia olelo ka mea, penei i ina o ka oiaio o ka manao like 'oiaʻiʻo a me ka piʻo' aoʻao o Ara, a crc.

Symmetry entails i ke ola ana o ka pilina ma, a ina o ka ka noʻonoʻo 'mea nana e hoole, oa he symmetrical pilina rearrangement o ka mea, aole ia i alakai i ka hou' ana o ka palapala "aeaie ana." No ka laʻana, i ka pilina o ka like o ka c mea symmetrical no i ka equivalence iwaena l = i; i like a¹s a me ka hoʻopono, e like me ka mea me ka launa pu, s¹a.

Transitive i - ia mea he waiwai ma i halawai pu me ke kēia koi: i au m, Z au y me ® Z au m, kahi ® hana like me ka hōʻailona ke kūapoʻana i nā hua'ōlelo: "ina ... laila ...". Ai 'haʻilula heluhelu, ke'ōlelo mai e like me: ". Inā kūʻokoʻa o ka m, Z no ka y me, Z like hana o ka m"