Prove that a+b=b+a

prove that a+b=b+a

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en.wikipedia.org/wiki/Proofs_involving_the_addition_of_natural_numbers#Proof_of_commutativity
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en.wikipedia.org/wiki/Proofs_involving_the_addition_of_natural_numbers#Proof_of_commutativity

>having to proove addition is addition
Has science gone too far?

The things that seem simple and obvious are often exactly the things people know least about e.g. all the subject matter people think is complicated / intellectual like advanced mathematics or chess are things AI can master relatively easily while all the subject matter people take for granted as "easy" or straightforward like being able to understand a joke or to conduct an artificial body's movements in playing a baseball game are way more complicated problems for AI to solve.

Here's your homework.

For which set?

Now do it with integers, rationals, reals and complex numbers.

I think the way you do it is just do it for the reals and you're good all the way down.

I kick you in the dick seven times, then I kick you in the dick nine times. Then tomorrow I kick you in the dick nine times, then I kick you in the dick seven times.
We'll repeat with different variations until we're both satisfied you understand how addition works.

What are a and b?

If they are real numbers, what definition did you use?

If we define the integers to be ordered pairs of naturals with the operations (a,b) + (c,d) = (a+c, b+d) and (a,b) * (c,d) = (ac+bd, bc + ad) with the equivalence relation (a,b) equivalent to (c,d) iff a+d = b + c then it is clear.

Moreover, it is clear for rationals by the construction of the quotient field.

If we define reals as equivalence classes of Cauchy sequences of rationals it is also clear.
It is also clear by construction if we use the Dedekind cuts contruction.

Not true in general case

This thread is retarded.

What are a and b? If they're elements of a group, or a ring, or a field, then associativity is an axiom. There is no proof.

You need to specify what a and b are. And define addition

youtu.be/iUrzicaiRLU

They follow immediately from the naturals, since their construction is based on them.

1+2=2+1

Check mate pepe.

That's not associativity, brainlet

a + ( - a) + b = b + a + ( -a)

b=b

Benis = Bagina

it's an axiom right?

Assume a and b are elements of an abelian additive group.

Check and mate.

Can you name a case where addition is not commutative?

Depends what you mean by addition.

If you define f+g to be the composite function of f and g, then it's clearly not commutative.

by associativity [math]a+b=\underbrace{(1+\cdots+1)}_{a}+\underbrace{(1+\cdots+1)}_{b}=\underbrace{(1+\cdots+1)}_{b}+\underbrace{(1+\cdots+1)}_{a}=b+a[/math]

Put 'a' pebbles in one group. Put 'b' pebbles in another group. Count all of the pebbles, starting with the first group and continuing with the second group. Then count all of the pebbles again, starting with the second group and continuing with the first group. Note whether you got the same count both times. Now repeat this experiment many times, with many different numbers of pebbles in the two groups. When you're satisfied that the counts always match, you can make an unproven but likely assumption that it works the same for any numbers of pebbles. Then you can call it an 'axiom,' and name it something impressive-sounding, and pretend that it's 'true,' and that it has nothing to do with physical reality at all, and that you invented it entirely within the realm of abstract thought. Then you can call yourself a mathematician.

Assuming a,b are complex numbers.

C forms a field, QED.

Ordinal arithmetic.