Is it Defined, Or Can It Be Proven?

[Bashyboy]

Can the equality a-b = a + (-b) be proven, or are a - b and a + (-b) defined to be the same?

 

[LCKurtz]

Can the equality a-b = a + (-b) be proven, or are a - b and a + (-b) defined to be the same?

It can be proven. I will use $^-b$ for the additive inverse of $b$ so not to confuse it with subtraction. So you want to know whether $a +^-b = a-b$. Remembering that $a-b$ is the number you can add to $b$ to get $a$, let's check whether it works:$$
b +(a + ^-b) = b +(^-b +a) = (b + ^-b) + a = 0 + a = a$$ so it works. Can you fill in the reason for each step?

 

[Bashyboy]

I understand each of the properties you appealed to, to justify each step; but I do not see how this shows that
a - b and a + (-b) are equivalent.

 

[LCKurtz]

I understand each of the properties you appealed to, to justify each step; but I do not see how this shows that
a - b and a + (-b) are equivalent.

I just showed that if you add $a+^-b$ to $b$ you get $a$. That is the definition of $a-b$ since $a-b$ is the number you can add to $b$ to get $a$. So $a+^-b$ is $a-b$.

 

[Fredrik]

I would take it as the definition of subtraction. If you want to prove it, you have to specify what other definition of subtraction that you're using. LCKurtz is defining a-b (for arbitrary a and b) as the unique number x such that b+x=a. If you add -b to both sides of this equality, you see that x=a+(-b). I wouldn't say that this is the definition of subtraction. It's just a definition of subtraction.

 

[D H]

Funny: The wikipedia article on subtraction uses LCKurtz's definition, but the wikipedia article on the integers uses Fredrik's definition.

With LCKurtz's definition, that subtraction is the inverse function of addition is axiomatic, but that subtraction is equivalent to adding with the additive inverse is a theorem. With Fredrik's definition, it's the other way around.

 

[thelema418]

Can the equality a-b = a + (-b) be proven, or are a - b and a + (-b) defined to be the same?

A little bit of both. The issue concerns whether (-b) is a unique element.

In abstract algebra the ring axiom says that for every $a$ there is some $x$ such that $a + x = 0_R$. It does not claim how many $x$ exist for a given $a$. So, this is what you have to prove.

If $-b$ isn't unique you might end up getting multiple answers when doing a subtraction!