Math proofs help

[Anonymous No.15218705]

Ik this question has been asked many times
Im taking my first proof based math class modern algebra and the concepts make sense if i put in the time to learn it
But I have no idea how to even start doing HW problems so i have to look them up

what does proving something even mean?
prove a(bc)=b(ac)
- abc=bac=x
x=x
proven? i'm not even sure what it is that I did because of course a(bc)=b(ac) that's just obvious. do I need to prove x=x too?

[Anonymous No.15218722]

>>15218705
>of course a(bc)=b(ac) that's just obvious
It isn't, though, especially if you're working with modern algebra.
If you're working with real numbers, complex numbers, or anything of the sort, you'd just need to prove/cite the commutative and associative properties, but there are plenty of contexts where this relationship isn't true at all

[Anonymous No.15218730]

>>15218705

It's just a way to practice how to reference axioms in proof as a stepping stone in referencing theorems in proofs. For example, you would say by associativity a(bc) = (ab)c and by commutivity for multiplication (ab)c = (ba)c , and (ba)c = b(ac) by associativity, therefore a(bc) = (ab)c.

[Anonymous No.15218732]

>>15218730
>>15218722

And yes, it probably stipulated that a,but, and c and real numbers or integers or something of the like. This wouldn't work for say matrices however.

[Anonymous No.15218739]

>>15218705
>prove a(bc)=b(ac)
unfortunately you need to prove it in terms you already have defined, or axioms (theories) everyone can agree on. In this case you "obviously" know they are equal, but you can't really explain why. Associativity and commutativity of multiplication can be proven through decomposition of multiplication into addition, that way you don't have to take anything for granted in your explanation (Except how addition works...).

[Anonymous No.15218759]

>>15218739
>>15218730
>>15218722
I need some practical advice for learning this stuff in general.
what's your thought process initially when asked to prove something?

[Anonymous No.15218766]

>>15218739
>>15218759
Proof is a very specific concept, I started to see it in Geometry class where'd you draw things with straightedge to prove things using parallel lines and equal angles. In order for something to be a concrete proof, it must rely entirely on mathematics which have already been proven (or at least very widely-accepted).

Here's one example; let's take the left side and try to manipulate it into being the right side, without using the distributive, commutative, or associative properties (because that's what you're asked to prove!).

a(bc) = a(b + b + ... + b) (c times)
= (a + a + ... + a) (b times) (c times)
= (ab) + (ab) + ... + (ab) (c times)
= (ab)c

This uses only the basic properties of addition and multiplication, no higher-order concepts. As long as we accept the rules of addition, what multiplication means in terms of addition, what the equals sign is; then we must also accept the commutative property.

[Anonymous No.15218769]

>>15218766
Sorry, I wrote this up wrong and proved a(bc)==(ab)c, instead of a(bc)==b(ac). But hopefully you can see, the process is identical.

[Anonymous No.15218790]

>>15218759
>what's your thought process initially when asked to prove something?
First try to think of obvious counterexamples or edge cases which would complicate or invalidate a proof. Those cases often clarify and pin down the boundaries of what is necessary to prove. For example, setting some values to zero, or reciprocals of each other, etc.

Recognizing which tools, principles, and methods are relevant to a proof, is what elevates mathematicians with knowledge and experience in their specific fields. A geometric proof is going to involve geometry, and possibly disciplines which overlap and relate to it (trigonometry, integral calculus, linear algebra...)