Ngā Mahi Whakauru, Ngā Mahi Tirotiro, me Ngā Mahi Whaiaro
I roto i te pāngarau, inā koa i roto i te ariā mahi, e toru ngā momo mahi nui e kōrerohia pinepinetia ana: te werohanga, te tirotirohanga, me te takiruahanga. He āhuatanga ahurei tō ia o ēnei momo mahi e toru e whakatau ana i te mahere o ngā huānga mai i te huinga pūtake (rohe) ki ngā huānga i roto i te huinga ūnga (awhe, rohe-ko rānei). Ka whakamāramahia e tēnei tuhinga te whakamāramatanga, ngā āhuatanga, me ngā tauira o ia o ēnei mahi, me ō rātou tono i roto i ngā mara rerekē.
Mahi Werohanga
Ko te mahi werowero, e mōhiotia ana ko te mahi kotahi-ki-te-kotahi, he mahi e honoa ana ia huānga o te huinga pūtake ki tētahi huānga ahurei o te huinga ūnga. I roto i te āhua ōkawa, ka kiia he mahi werowero te mahi \( f : A \to B \) mēnā, ā, mēnā anake mō ia \( a_1, a_2 \in A \), \( f(a_1) = f(a_2) \) e tohu ana ko \( a_1 = a_2 \).
He māmā ake te whakaaro, mā te mahi werohanga ka kore e rua ngā huānga motuhake o te huinga pūtake e whai ahua ōrite ana i roto i te huinga ūnga. Arā, kotahi te huānga pūtake o ia huānga o te huinga ūnga e hono ana ki taua huānga.
Tauira:
– Whakaarohia te mahi \( f: \mathbb{R} \to \mathbb{R} \) kua tautuhia ko \( f(x) = 2x + 3 \). He mahi werohanga tēnei nā te mea mēnā \( f(a) = f(b) \), kātahi \( 2a + 3 = 2b + 3 \), ko te tikanga \( a = b \).
Taupānga:
He maha ngā wā ka whakamahia ngā mahi werohanga i ngā horopaki e hiahia ana tātou kia kore ai he tāruarua, pērā i te tātaitanga, te waehere rānei.
Mahi Whakamātautau
Ko te mahi tirotiro, te mahi-ki-runga rānei, he mahi kei roto i ia huānga o te huinga ūnga \( B \) he huānga kotahi pea te iti rawa mai i te huinga pūtake \( A \) e hono ana ki a ia. I roto i te tuhi ōkawa, ka kiia he mahi tirotiro he mahi \( f : A \ki B \) mēnā mō ia \( b \in B \), kei reira te iti rawa kotahi \( a \in A \) kia \( f(a) = b \).
Arā, mā te mahi tātaritanga ka kapi katoa te huinga ūnga e te ahua o te huinga pūtake. Kāore he huānga o te huinga ūnga e "kapia".
Tauira:
– Whakaarohia te mahi \( f: \mathbb{R} \to \mathbb{R} \) kua tautuhia ko \( f(x) = x^3 \). He mahi whakaaroaro tēnei nā te mea mō ia \( y \in \mathbb{R} \), ka kitea e tātou \( x \in \mathbb{R} \) kia \( x^3 = y \).
Taupānga:
E whakamahia whānuitia ana ngā mahi tātari i roto i te horopaki o te tohatoha rauemi, me whakarite kia whiwhi ia kaiwhiwhi i tētahi mea mai i te huinga o ngā kaituku.
Pānga Tauira
Ko te mahi takirua he mahi werohanga, he mahi tirotiro hoki. Arā, ko te mahi takirua he mahi takitahi-ki-te-kotahi, ā, he mahi ki runga hoki. Nō reira, i roto i te mahi takirua, ka maherehia motuhaketia ia huānga o te huinga pūtake ki tētahi huānga o te huinga ūnga, ā, i tētahi atu taha, kotahi tonu te huānga o ia huānga o te huinga ūnga e maherehia ana ki a ia mai i te huinga pūtake.
Tauira:
– Whakaarohia te mahi \( f: \mathbb{R} \to \mathbb{R} \) kua tautuhia ko \( f(x) = x + 1 \). He mahi takirua tēnei nā te mea:
– Whakatakotoranga: Mena ko te \( f(a) = f(b) \), ko te \( a + 1 = b + 1 \), e tohu ana i te \( a = b \).
– Arotakenga: Mō ia \( y \in \mathbb{R} \), ka kitea e tātou \( x = y – 1 \) kia \( f(x) = y \).
Taupānga:
He mea tino nui ngā mahi takirua i roto i te horopaki o ngā panonitanga me ngā isomorphisms, ina hiahiatia kia tiakina te hanganga, ngā whanaungatanga rānei i waenga i ngā huānga ina maherehia mai i tētahi huinga ki tētahi atu. Hei tauira, i roto i te whakamunatanga, he maha ngā wā he mahi takirua ngā kī whakamunatanga me te wetemunatanga kia taea ai te whakamuna me te wetemuna motuhake i ngā karere.
Tātaritanga Anō
Ngā Whakairoiro me ngā Kauwhata
He mea āwhina te whakamahi i tētahi tūtohi Venn, i tētahi kauwhata rānei, kia mārama ai ki ēnei mahi. I roto i tētahi tūtohi Venn, ka taea te whakaatu i tētahi mahi werohanga mā ia huānga o te huinga ūnga me te kotahi te pere e haere mai ana. Ka taea te whakaatu i tētahi mahi tirotiro mā ia huānga o te huinga ūnga me te kotahi te pere e haere mai ana. He mahi takirua kei ia huānga o ngā huinga pūtake me ngā huinga ūnga he pere e haere mai ana, ka hangaia he taurite kotahi-ki-te-kotahi.
Pānga Whakamuri
Ko tētahi atu āhuatanga nui e akohia ana i roto i te horopaki o ngā mahi werohanga, mahi tirotiro, me ngā mahi takirua ko te mahi whakamuri.
– He mahi whakamuri maui tonu tā te mahi werohanga.
– He mahi whakamuri matau tonu tā te mahi tātaritanga.
– He mahi whakamuri ahurei tonu tā te mahi tauwhāiti.
Mena he pānga taha rua tētahi mahi, ka noho ngā whakahurihuri maui me ngā whakahurihuri matau, ā, ka ōrite ngā mea e rua, ka hangaia te mahi whakahurihuri tūturu.
Te Katinga
He mea nui te mārama ki ngā ariā o ngā mahi werohanga, te mahi tirotiro, me te mahi takirua ki ngā manga maha o te pāngarau me ā rātou tono mahi. Mā ngā mahi werohanga ka kore e tāruarua; mā ngā mahi tirotiro ka kapi katoa; ā, mā ngā mahi takirua ka taurite te hononga takitahi i waenga i ngā huānga i roto i ngā huinga e rua. He mea nui te mōhio ki ēnei momo mahi e toru, ehara i te mea i roto i te pāngarau parakore anake, engari i roto hoki i ngā mara pērā i te pūtaiao rorohiko, te ōhanga, me te hangarau. Mā te māramatanga hōhonu ki ngā mahi me ngā tono o ēnei mahi ka taea te whakatuwhera i te kuaha ki te tātari me te whakaoti rapanga whai hua me te whai hua.
