Let m, n e N. If n >= m, then there is a unique b e N such that n = m + b.

How do I prove that b is unique.

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- Feb 16th 2009, 08:50 PMjzellt[SOLVED] Proof: uniqueness of a natural number
Let m, n e N. If n >= m, then there is a unique b e N such that n = m + b.

How do I prove that b is unique. - Feb 16th 2009, 08:56 PMChris L T521

$\displaystyle m,n\in\mathbb{N},n\geqslant m\implies\exists!\,b\in\mathbb{N}:n=m+b$

Suppose not. Let $\displaystyle m,n\in\mathbb{N}$ and let $\displaystyle b,c\in\mathbb{N}$ such that $\displaystyle n=m+b$ and $\displaystyle n=m+c$. By transitivity, we see that $\displaystyle m+b=m+c$. Thus, by cancellation, we have $\displaystyle b=c$. This shows that $\displaystyle b$ is unique.

Does this make sense? - Feb 16th 2009, 08:57 PMJhevon