Associative vs Commutative
Nyob hauv peb lub neej niaj hnub, peb yuav tsum siv cov lej thaum twg peb xav tau qhov ntsuas ntawm ib yam dab tsi. Ntawm lub khw muag khoom noj, ntawm qhov chaw nres tsheb roj, thiab txawm nyob hauv chav ua noj, peb yuav tsum tau ntxiv, rho tawm, thiab muab ob lossis ntau qhov ntau ntxiv. Los ntawm peb qhov kev xyaum, peb ua cov kev suav no tsis muaj zog. Peb yeej tsis pom lossis nug vim li cas peb ua cov haujlwm no hauv txoj kev tshwj xeeb no. Los yog vim li cas cov kev suav no tsis tuaj yeem ua rau lwm txoj hauv kev. Cov lus teb tau muab zais rau hauv txoj kev cov haujlwm no tau txhais hauv kev ua lej ntawm algebra.
Nyob hauv algebra, kev ua haujlwm uas muaj ob qhov ntau (xws li ntxiv) txhais tau tias yog kev ua haujlwm binary. Ntau precisely nws yog ib tug ua hauj lwm ntawm ob lub ntsiab los ntawm ib tug teeb thiab cov ntsiab lus no hu ua 'operand'. Ntau qhov haujlwm hauv lej suav nrog kev ua lej lej uas tau hais ua ntej thiab cov uas tau ntsib nyob rau hauv txoj kev xav, linear algebra, thiab lej logic tuaj yeem txhais tau tias yog kev ua haujlwm binary.
Muaj ib txoj cai tswjfwm ntsig txog kev ua haujlwm binary tshwj xeeb. Associative thiab commutative properties yog ob qho tseem ceeb ntawm kev ua haujlwm binary.
Ntau txog Commutative Property
Xav tias qee qhov kev ua haujlwm binary, qhia los ntawm lub cim ⊗, ua tiav ntawm cov ntsiab lus A thiab B. Yog tias qhov kev txiav txim ntawm cov operands tsis cuam tshuam rau qhov tshwm sim ntawm kev ua haujlwm, ces kev ua haujlwm tau hais tias yog kev sib hloov. i.e. yog tias A ⊗ B=B ⊗ A ces kev khiav hauj lwm yog commutative.
Kev ua lej lej sib ntxiv thiab sib npaug yog sib piv. Qhov kev txiav txim ntawm cov lej ntxiv ua ke lossis sib npaug ua ke tsis cuam tshuam rau cov lus teb kawg:
A + B=B + A ⇒ 4 + 5=5 + 4=9
A × B=B × A ⇒ 4 × 5=5 × 4=20
Tab sis nyob rau hauv cov ntaub ntawv ntawm kev faib hloov nyob rau hauv qhov kev txiav txim muab lub reciprocal ntawm lwm tus, thiab nyob rau hauv rho tawm qhov kev hloov muab qhov tsis zoo ntawm lwm tus. Yog li ntawd, A – B ≠ B – A ⇒ 4 – 5=-1 and 5 – 4=1
A ÷ B ≠ B ÷ A ⇒ 4 ÷ 5=0.8 thiab 5 ÷ 4=1.25 [qhov no A, B ≠ 1 thiab 0
Qhov tseeb, qhov rho tawm tau hais tias yog kev tiv thaiv kev sib txuas; qhov twg A – B=– (B – A).
Tsis tas li, kev sib txuas lus sib txuas, kev sib txuas, kev sib cais, kev cuam tshuam, thiab qhov sib npaug, kuj yog kev sib txuas. Qhov tseeb muaj nuj nqi kuj yog kev sib hloov. Cov txheej txheem ua haujlwm union thiab kev sib tshuam yog kev sib hloov. Ntxiv thiab cov khoom scalar ntawm cov vectors kuj yog commutative.
Tab sis cov vector rho tawm thiab cov khoom vector tsis sib txuas (vector khoom ntawm ob vectors yog anti-commutative). Qhov sib ntxiv ntawm matrix yog commutative, tab sis qhov sib npaug thiab qhov rho tawm tsis yog sib piv.(Multiplication of two matrices can be commutative in special case, such as the multiplication of a matrix with nws inverse or the identity matrix; but mas matrices is not commutative if the matrices not of the same size)
Ntau txog Associative Property
Kev ua haujlwm binary tau hais tias yog kev sib koom ua ke yog tias qhov kev txiav txim ntawm kev ua tiav tsis cuam tshuam rau qhov tshwm sim thaum muaj ob lossis ntau qhov tshwm sim ntawm tus neeg teb xov tooj tam sim no. Xav txog cov ntsiab lus A, B thiab C thiab kev ua haujlwm binary ⊗. Kev ua haujlwm ⊗ tau hais tias yog koom nrog yog
A ⊗ B ⊗ C=A ⊗ (B ⊗ C)=(A ⊗ B) ⊗ C
Los ntawm cov lej lej yooj yim, tsuas yog qhov sib ntxiv thiab qhov sib npaug yog qhov sib koom.
A + (B + C)=(A + B) + C ⇒ 4 + (5 + 3)=(5 + 4) + 3=12
A × (B × C)=(A × B) × C ⇒ 4 × (5 × 3)=(5 × 4) ×3=60
Kev rho tawm thiab faib tsis yog koom nrog;
A – (B – C) ≠ (A – B) – C ⇒ 4 – (5 – 3)=2 and (5 – 4) – 3=-2
A ÷ (B ÷ C) ≠ (A ÷ B) ÷ C ⇒ 4 ÷ (5 ÷ 3)=2.4 thiab (5 ÷ 4) ÷ 3=0.2666
Qhov kev sib txuas lus sib txuas sib txuas, sib txuas, thiab sib npaug yog sib koom ua ke, xws li kev teeb tsa kev sib koom ua ke thiab kev sib tshuam. Lub matrix thiab vector ntxiv yog associative. Cov khoom scalar ntawm vectors yog associative, tab sis cov khoom vector tsis yog. Matrix multiplication yog associative tsuas yog nyob rau hauv cov xwm txheej tshwj xeeb.
Qhov txawv ntawm Commutative thiab Associative Property yog dab tsi?
• Ob qho khoom sib koom ua ke thiab cov cuab yeej sib txuas yog cov khoom tshwj xeeb ntawm kev ua haujlwm binary, thiab qee qhov txaus siab rau lawv thiab qee qhov tsis ua.
• Cov khoom no tuaj yeem pom nyob rau hauv ntau hom kev ua haujlwm algebraic thiab lwm yam haujlwm binary hauv lej, xws li kev sib tshuam thiab kev sib koom ua ke hauv kev teeb tsa kev xav lossis kev sib txuas lus.
• Qhov sib txawv ntawm kev sib txuas lus thiab kev sib koom ua ke yog cov khoom sib txuas lus hais tias qhov kev txiav txim ntawm cov ntsiab lus tsis hloov qhov kawg tshwm sim thaum cov cuab yeej koom nrog hais, tias qhov kev txiav txim hauv kev ua haujlwm, tsis cuam tshuam rau cov lus teb kawg..
