Maths

Make sure what your proving is countable/finite.
Step 1: Prove the first case (base case)
Step 2: Assume that your hypothesis holds, prove your hypothesis holds for the next step.

Done.

>2012+4
>using induction
I sincerely hope you don't do this

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>>7942860
No lmao you need the index set to be well ordered. You can't do induction on Z or Q.

>>7942904
are you saying Z and Q cant be well-ordered?

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>>7942934
They literally aren't though lmao.
>conflating well-ordering with total-ordering
Stay in school kids.

>>7942953
so just to be clear, you think there's only one possible way to order a given set? or were you just pretending

>>7942963
Give me any total order on Z and I'll show you it's not well ordered with respect to it

>>7942965
"The following relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds:

x = 0
x is positive, and y is negative
x and y are both positive, and x ≤ y
x and y are both negative, and |x| ≤ |y|
This relation R can be visualized as follows:

0 1 2 3 4 ... −1 −2 −3 ... "

>>7942967
Who are your quoting?
>imposing order after infinitely many numbers
Lmao ok.

>>7942975
wikipedia

>>7942860
What kind of baby only inducts on countable sets?

>>7942982
>transfinitists
Fuck the whole [math]\omega^\omega [/math] lot of you

Why cant you induct over the nonnegative reals?

for example

proof all non negative real numbers are non negative:

1: 0 >= 0
2: n >= 0
3: n + dn >= 0 + dn >= 0

>>7942831
What Math class is this? W..we don't do proofs in calc 2 and my cc.

>>7943040
It's called Basic Mathematics, obligatory course for anyone going into engineering or sciences. It is taught on your first semester.

Let A(n) be the statement you want to prove in respect to n, where n is an integer. First of all, you need to find a case where A is true for a particular n. Usually this is A(0) or A(1). If there's no such case, the proof is not possible. After this, you need to assume that the statement you want to prove is true. Now show that, if A(n) is true, then also A(n+1) is true. That's usually the challenging part. Because you usually have to prove (in)equations or divisibility relations, it is recommendable to start by writing out one side of the (in)equation respectively the term left to the | sign. Keep in mind that you have to write out these terms already with n+1. Then you have to transform the term in a way that the term of A(n) is detectable. Now you can replace this term by the respective left- or right-hand side of A(n). After that, you have to rearrange the term again until the term equals the left- or right-hand side respectively the term right to the | sign depending on what side you've begun with. If you were capable of achieving this, your proof is finished. The process of rearranging the terms may involve any conceivable sort of algebraic transformation, so you need to be creative and familiar with these "tricks". You may never lose the goal - the other side of the (in)equation respectively addends that are divisible by the term right to the | sign - out of sight. All in all, the principle of proofs by induction is that you find one integer for which your statement is true. Afterwards you show that if the statement is true, then the statement is true for the subsequent integer and since you've found one for which the statement is true, it is also true for the subsequent integer. Now you've found another one for which it is true which implies that it is true for the subsequent integer and so on. To conclude, the statement holds - after finishing the proof correctly - for all natural numbers equal to or bigger than the integer you have found in the beginning.

for some reason, one thing that i didn't even think about doing until abstract algebra was using a number other than 1 as the increment in the inductive step
you can use this to prove things about even numbers and multiples of primes and whatnot
you can even use negative numbers

OP, when I first learned induction long ago, something I struggled with was not necessarily knowing what the final step should look like, or is supposed to look like, if the thing you're being asked to prove is in fact true. And so I flailed around, not knowing what direction to manipulate the expressions in.

Here is a valid technique that I have personally found very useful for first learning induction proofs: Write the beginning steps at one end of your paper (specifically inductive hypothesis), /and then figure out what the final step is supposed to look like, and write that out at the opposite end of your paper/. The latter will (generally) just be the exact format that the thing takes in the (n+1) case, as opposed to the (n) case.

NOW. Your task is to show that the two things on either end of the paper are in fact equal --- you haven't proven this yet, and that's the whole point. To accomplish this, since you now have two different things that you can actually compare and look at, it should in most cases be fairly simple to manipulate them closer together, until eventually they meet in the middle. Having two things to compare now, you'll actually be better able to visually scan and tell if you're working in the right direction. You are literally working the problem from both ends.

cont.

>>7945117

IF, somewhere in the middle, you can conclude that the two things ARE in fact equal (or whatever the appropriate relation is), THEN you will be able to conclude your original thing (for all n, P(n), etc etc). However, IF after very careful manipulation you are able to conclude that the two things are definitely NOT algebraically equal, THEN you will obliged to conclude that the appropriate statement is false. And in this case, you can probably cook up a concrete counterexample, maybe further along from the basis case that you checked (or perhaps there had been a mistake with the basis case itself), to convince yourself more easily that the statement is in fact false.

The above process can be messy at first, so allow yourself time and space to do clean-up if need be, on a homework or test. I recommend blank printer paper. Once you've done this a few times, you'll have a much better sense of how these proofs are supposed to go, and where you're supposed to end up, so that you won't have to do the above anymore.

>>7943031
Real numbers are uncountably infinite, the basis of induction is to show how the thing you're proving can be proven by showing that the (countably infinite) natural numbers have no upper bound

>>7945158
>the (countably infinite) natural numbers have no upper bound
neither does the reals

>>7942831
> Proof by induction is the first topic in my college maths
Lolwhat? Where are you from? Isn't this stuff supposed to be covered in school?

>>7945228

This is the second time today I've noticed "school" being used in a confusing hs/college/uni context.

Get it straight. It's all 'school', from kindergarten to postdoc and beyond. 'school' doesn't stop at a given country's diploma ceremony for its equivalent of high school. Because the whole bit is: education, particularly formal educational systems, a.k.a. school.