abstract algebra - Proving distributive law in $\mathbb{Z}_{n}$ for equivalence classes - Mathematics Stack Exchange


equivalence classes defined $\overline{a} = \{ b \in \mathbb{z} : b \equiv \mod n \}$ .

i need prove distributive law in $\mathbb{z}_{n}$. so, if $\overline{a}$, $\overline{b}$, , $\overline{c}$ arbitrary elements in $\mathbb{z}_{n}$, $\overline{a}*(\overline{b}+\overline{c}) = \overline{a}*\overline{b}+\overline{a}*\overline{c}$.

i've seen examples proving distributive law real numbers , understood, i'm not sure if same proof works equivalence classes.

the definition of operations on $\mathbb{z}_n$ $$ \bar{a}+\bar{b}=\overline{a+b} \qquad \bar{a}\,\bar{b}=\overline{ab} $$ $$ \bar{a}(\bar{b}+\bar{c})= \bar{a}\overline{(b+c)}= \overline{a(b+c)}=\overline{ab+ac} $$ can finish?


Comments

Post a Comment

Popular posts from this blog

analysis of two transistors ib,ic,ie? - Electrical Engineering Stack Exchange

Image
is ib current correct went kvl through first transistor your forgetting re. you need use effective resistances r4 , r6, value \$r4_e = r4 * (1+\beta) = 40,000 * 101 = 4,040,000\omega\$ \$r6_e = r6 * (1+\beta) = 1,000 * 101 = 101,000\omega\$ using base circuit looks this. simulate circuit – schematic created using circuitlab you solve using various methods, shortcut realise that, ideal vbe, voltage @ top of both emitter resistors identical. such can treat both legs being in parallel. circuit can further simplified to.. simulate circuit now should able calculate current through circuit as.. \$i_{rb} = (2.6 - 0.6)/(2,260 + 98,540) = 19.84ua\$ that makes base voltages both.. \$v_b = 2.6 - 19.84ua * 2.26k\omega = 2.555v\$ so \$ib1 = (2.555-0.6)/4.04m\omega = 0.484ua\$.. , \$ib2 = (2.555-0.6)/101k\omega = 19.356ua\$ however: these numbers make sense if transistor not saturated. question states \$i_e \approx \beta * i_b\$ indicates ...
Read more

Choosing and replace capacitors for audio amplifier - Electrical Engineering Stack Exchange

Image
i have old audio amplifier (jvc a-gx2b) produces crackling noises on 1 of outputs. so, opened , discovered 2 largest capacitors seem leaking : can replace capacitor same characteristics (4700uf 50v), or should instead take specific capacitors audio? if yes, ones ? in addition, have interest replace capacitors of amp (not 2 largest)? 1: wear safety gloves when handling toxic leaky capacitors, remove , clean safely isoprop. , mark note polarity (-) 2 : measure dia , lead pitch dxl [mm] sort parts this 3: audio or better low esr, highest ripple current caps. < 50mohm, > 3aripple 4: better caps run cooler lower esr , rated higher temp. may not fit. 5: japanese caps have better reliability i suggest 1st choice panasonic @ digikey .. cost of shipping may more cost of caps. as suggested: cap voltage must same or more - capacitance tolerance doesn't matter, , may larger value. - large e-caps may replaced @ same time.
Read more

sql server 2008 r2 - LogWriter: Operating system error 1117 - Database Administrators Stack Exchange

i getting these errors on dedicated server because of database goes recovery mode. i have seen following error in sql server error log file: 2018-01-20 05:32:19.83 spid7s logwriter: operating system error 1117(the request not performed because of i/o device error.) encountered. 2018-01-20 05:32:19.83 spid7s write error during log flush. 2018-01-20 05:32:19.84 spid284 client unable reuse session spid 284, had been reset connection pooling. failure id 46. error may have been caused earlier operation failing. check error logs failed operations before error message. 2018-01-20 05:32:19.84 spid335 database newtrack shutdown due error 9001 in routine 'xdesrmfull::commit'. restart non-snapshot databases attempted after connections database aborted. 2018-01-20 05:35:03.35 spid22s recovery completed database aa(database id 7) in 151 second(s) (analysis 18856 ms, redo 115386 ms, undo 2023 ms.) informational message only. no user action required. at same ti...
Read more