1

I was wondering is it possible to prove 1+0=1(or 0+1=1)? If yes! how does it look like! Does it even make sense to ask for proof here? My google searches didn't turn up anything.

Thanks.

Ashish Shukla
  • 157
  • 2
    The former is basically the definition of '+' and '0'. The latter also needs the definition of '1'. Look up Peano axioms. But yeah, in some sense this cannot be proved. – Jyrki Lahtonen Sep 23 '19 at 06:39
  • 1
    You can only prove things within a system with defined axioms an definitions. In nearly all mathematical systems the definition of $0$ is that $0$ is the additive identity; that is $0$ is defined to be a number so that $a+0 = 0 + a = a$ always. $0+1=1$ by definition. But it may vary for different systems. – fleablood Sep 23 '19 at 06:53
  • 1
    If you can tell me how you defined "+" and $0$, I can tell you if $0+1 = 1$ is something to prove or something defined by axiom. Jyrki Lahtonen makes an interesting comment that in the Peano postulates that $a+0=a$ is axiomatic but you must prove that $0+a=a$ (and that $a+b = b+a$). In group theory $a+0=0+a = a$ is an axiom. – fleablood Sep 23 '19 at 06:59
  • 2
    In Peano arithmetic you start with things like $x+0=x$ and $x+S(y)=S(x+y)$ and $1=S(0)$. The first of these gives $1+0=1$ immediately, while combined they give $0+1=0+S(0)=S(0+0)=S(0)=1$ – Henry Sep 23 '19 at 07:26

1 Answers1

0

$x+0=0+x=x$ is an axiom in Peano arithmetic's implicit definition of the addition of non-negative integers. We set $x=1$ for the proof.

J.G.
  • 115,835
  • 6
    When I learned Peano axioms $x+0=x$ was true by definition. On the other hand $0+x=x$ required a proof by induction on $x$. – Jyrki Lahtonen Sep 23 '19 at 06:42
  • @JyrkiLahtonen Typically either $0+x$ or $x+0$ will be mentioned in the definition; the rest follows when you prove commutativity. – J.G. Sep 23 '19 at 07:28