how would you approach task 10a?
translated:
10. In this task, n is a fixed, natural number. if (…) we write (…) if a - b is divisible by n.
a) show that if (…) and (…), then (…).
(to see what “(…)” is look at the picture below)
#calculus
ionic idol
proper fjordBOT
how would you approach task 10a?
translated:
10. In this task, n is a fixed, nat...
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ionic idol
@upbeat forum do you have any idea?
pearl raven
how would you approach task 10a?
translated:
10. In this task, n is a fixed, nat...
Just use the definitions of $a\equiv b$ and $b\equiv c$ to show there exists integer $k$ such that $a-c=nk$, to which by definition $a\equiv c$
thin reefBOT
Omegabet_
upbeat forum
Modular huh
Well $a \equiv b (\mod n)$ (n is your fixed constant here) basically means that the remainder when both $a$ and $b$ are divided by $n$ are the same
thin reefBOT
Wawi #NwoWifer
upbeat forum
So, you can write $a=ln+k$ and $b=mn+k$
$k,l,m\in\bN$
(And btw this isn’t calculus, this is number theory)
thin reefBOT
Wawi #NwoWifer
ionic idol
(And btw this isn’t calculus, this is number theory)
oh you are right, my bad
but what does it mean when a number is fixed
like here, n
thin reefBOT
septembry
ionic idol
but just if a and b is an element of integres
pearl raven
but what does it mean when a number is fixed
n just represents an integer that wont change
yes, $a\equiv b$ is defined to be the statement '$a-b$ divides $n$'.
thin reefBOT
Omegabet_
ionic idol
understood, thanks
ionic idol
**Omegabet\_**
i see it now, but how do i show it?
pearl raven
i see it now, but how do i show it?
Like I said... use the definitions
a\equiv b means a-b=nm for some integer m
b\equiv c means ...
Use those to show a-c=nk for some integer k
ionic idol
**Omegabet\_**
well i didnt include k, but would it be the “same”?
i used x and y instead
pearl raven
i used x and y instead
You're working over the integers, so just work with integers
a-b=nx
b-c=ny
a-c=a-b+b-c=nx+ny=n(x+y), and x+y is an integer
So yes, right idea, but it's sporadic as a proof (thus hard to read)
ionic idol
So yes, right idea, but it's sporadic as a proof (thus hard to read)
aaa ok
ionic idol
You're working over the integers, so just work with integers
but how did i work over the integers?
pearl raven
a-b=nx only uses integers
Once you divide by n, you're introducing the rationals
Since 1/n isn't an integer (unless n=+-1)
ionic idol
Once you divide by n, you're introducing the rationals
ohhh i see
pearl raven
It's not wrong perse to write x=(a-b)/n, just not proper since this is an equivalence relation over Z
Hence only facts about Z should be used
ionic idol
Once you divide by n, you're introducing the rationals
but how did i divide it?
i just left a and b alone on the other side
ionic idol
idk what else i should have done
ionic idol
**Omegabet\_**
well i could have used this maybe
ionic idol
It's not wrong perse to write x=(a-b)/n, just not proper since this is an equiva...
wait what does it mean when its an equivalent relation (over Z)?
ionic idol
Hence only facts about Z should be used
what do you mean by “facts about Z should be used”
pearl raven
wait what does it mean when its an equivalent relation (over Z)?
An equivalence relation is a relation that is reflexive, symmetric, and transitive
Facts as in properties of Z
pearl raven
idk what else i should have done
But again, your proof is correct... just rough looking
ionic idol
But again, your proof is correct... just rough looking
well thank you so much!
surreal summitBOT
@ionic idol has given 1 rep to @pearl raven
ionic idol
+close