Ayyuka da Marasa Aiki: Fahimtar Ra'ayoyi a Lissafi
A fannin lissafi, sau da yawa muna cin karo da ra'ayoyi marasa ma'ana waɗanda ke buƙatar zurfafa fahimta. Ɗaya daga cikin irin wannan muhimmin ra'ayi shine "aikin." Wannan ra'ayi ba wai kawai yana da rinjaye a cikin lissafi ba, har ma yana samuwa a fannoni daban-daban kamar kimiyyar lissafi, kimiyyar kwamfuta, da tattalin arziki. Duk da haka, rikice-rikice yakan taso lokacin da ake bambance tsakanin ayyuka da waɗanda ba ayyuka ba. Wannan labarin zai tattauna dalla-dalla game da menene aiki, halayensa, da kuma yadda za a bambanta shi da waɗanda ba ayyuka ba.
Menene Aiki?
A taƙaice dai, aiki dangantaka ce da ke haɗa kowane abu a cikin saiti ɗaya da daidai abu ɗaya a cikin wani saitin. Bari mu dubi ma'anarsa ta asali:
Ma'anar Aiki ta Asali:
Aikin f daga saitin A zuwa saitin B ƙa'ida ce da ke haɗa kowane abu a cikin A da daidai abu ɗaya a cikin B. An rubuta shi kamar haka \( f : A \rightarrow B \).
A wata ma'anar, idan muna da abubuwa daban-daban guda biyu a cikin saitin A, dole ne su kasance ba su da alaƙa da abu ɗaya a cikin saitin B.
Misali mai sauƙi na aiki shine zafin jikin mutum (yanki/saitin A) da kuma ma'auninsa a digiri Celsius (kewayon/saitin B). Kowane mutum yana da alaƙa da ƙimar zafin jiki ɗaya. Don haka, a cikin wannan misalin, aikin yana danganta mutane (abubuwan yanki) da ƙimar zafin jiki (abubuwan yanki).
Halayen Aiki
A fahimtar wannan ra'ayi, akwai wasu takamaiman halaye da dole ne dangantaka ta cika domin a ɗauke ta a matsayin aiki:
1. Alaƙa tsakanin saitin:
Dole ne aikin ya haɗa kowane abu a cikin yankin (saitin tushe) da daidai abu ɗaya a cikin yankin haɗin gwiwa (saitin manufa). Idan wani abu a cikin yankin yana da alaƙa da abubuwa fiye da ɗaya a cikin yankin haɗin gwiwa, to ba aiki bane.
2. Bambanci:
Za a iya danganta wani abu a cikin yankin da ƙima ɗaya kawai a cikin yankin haɗin gwiwa. Misali, idan f(a) = 3 da f(a) = 5 a lokaci guda, to f ba aiki bane.
3. Yankin Ba Komai Ba Ne:
A cikin aiki, kowane abu a cikin yankin dole ne ya sami takwara a cikin yankin haɗin gwiwa, ma'ana yankin ba zai iya zama fanko ba.
Ba Aiki Ba
Ana cewa dangantaka da ba ta cika halaye uku da ke sama ba ba ta aiki. Domin fahimtar wannan sosai, bari mu bincika wasu misalai na zahiri.
1. An Haɗa Siffa Ɗaya a cikin Yankin da Siffa Fiye da Ɗaya a cikin Yankin Haɗin gwiwa:
Misali, ka yi tunanin muna da dangantaka inda ɗalibai (yanki) ke da alaƙa da sakamakon jarrabawarsu (codomain). Idan ɗalibi zai iya samun maki fiye da ɗaya a jarrabawa ɗaya (misali, 75 da 90 a jarrabawa ɗaya), wannan dangantaka ba aiki ba ce.
2. Ba Duk Abubuwan Yankin Suna da Biyu Ba:
Kamar yadda aka ambata, a cikin aikin, dukkan abubuwan da ke cikin yankin dole ne su sami abokin tarayya a cikin yankin haɗin gwiwa. Idan wani abu ba shi da abokin tarayya, to dangantakar ba aiki ba ce.
Me yasa Fahimtar Aiki yake da Muhimmanci?
Fahimtar waɗannan ayyuka ba wai kawai yana da mahimmanci a cikin ilmin lissafi ba, har ma a aikace-aikace a wasu fannoni daban-daban:
1. Kimiyyar Kwamfuta:
Ayyuka su ne ginshiƙin shirye-shirye, kayan aikin da muke amfani da su don gina tsarin algorithms. Aiwatar da ayyuka yana ba mu damar rubuta lambar da ta fi sauƙi kuma mai sauƙin fahimta.
2. Ilimin kimiyyar lissafi:
Ana bayyana yawancin ma'aunin yanayi da dokokin zahiri ta hanyar ayyuka. Misali, dokar sanyaya iska ta Newton za a iya bayyana ta a matsayin aikin zafin jiki da lokaci.
3. Tattalin Arziki:
Ana amfani da ayyuka a cikin samfuran tattalin arziki don annabta alaƙar da ke tsakanin masu canji daban-daban kamar samun kuɗi da amfani.
Wakiltar Aiki
Ana iya wakiltar ayyuka ta hanyoyi daban-daban, dangane da buƙatu da mahallin:
1. Tebur:
Ana iya wakiltar ayyuka ta hanyar tebura da ke nuna nau'ikan abubuwan yanki da na yanki. Misali, teburi da ke nuna yanayin zafin yau da kullun (yanki) da adadin mazubin ice cream da aka sayar (yanki).
2. Jadawali:
Jadawalin aiki suna ɗaya daga cikin hanyoyin da suka fi tasiri wajen wakiltar ayyuka. A kan jadawali, galibi ana nuna yankin ta hanyar axis x da kuma haɗin kai ta hanyar axis y. Aiki jadawali ne da ya wuce gwajin layi na tsaye - wato, kowane layi na tsaye yana haɗuwa da jadawali a daidai wuri ɗaya.
3. Daidaito:
Siffar aljabra ta aikin ita ce hanya mafi yawan gaske a fannin lissafi don bayyana aiki. Misali, aikin layi \( f(x) = mx + c \) ko aikin kwata \( f(x) = ax^2 + bx + c \).
Misalan Ayyuka da Abubuwan da Ba Su Dace da Aiki Ba
Bari mu dubi wasu misalai don fahimtarmu game da ayyuka:
Misali na 1:
– Aiki: Muna da saitin A = {1, 2, 3} da saitin B = {4, 5, 6}. Aikin \( f \) yana bayyana cewa \( f(1) = 4 \), \( f(2) = 5 \), da \( f(3) = 6 \).
– Ba Aiki Ba: Idan \( f \) ya bayyana cewa \( f(1) = 4 \) da \( f(1) = 5 \), wannan dangantaka a bayyane take ba aiki ba ne saboda abu ɗaya a cikin yankin yana da alaƙa da abubuwa fiye da ɗaya a cikin yankin haɗin gwiwa.
Misali na 2:
– Aiki: Ana danganta ɗalibi (yanki) da sakamakon gwajinsa (codomain).
– Ba Aiki Ba: Ɗalibi ɗaya ba zai iya samun maki biyu daban-daban a jarrabawa ɗaya ba. Idan akwai maki biyu, dangantakar ba aiki ba ce.
Kammalawa
Idan aka yi la'akari da bayanin da ke sama, za mu iya ganin cewa aiki wani muhimmin ra'ayi ne a fannin lissafi wanda ke samun aikace-aikace da yawa a fannoni daban-daban. Aiki yana haɗa kowane abu a cikin saitin yanki tare da daidai abu ɗaya a cikin saitin yanki. Idan dangantaka ta kasa biyan waɗannan halaye, ba aiki bane. Fahimtar bambanci tsakanin aiki da rashin aiki yana ba da tushe mai ƙarfi don ƙarin nazarin lissafi da aikace-aikacen aiki a rayuwar yau da kullun.
