Properties of Affine Spaces

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Theorem

Let $\mathcal E$ be an affine space with difference space $V$.

Let $0$ denote the zero element of $V$.

Then the following hold for all $p,q,r \in \mathcal E$ and all $u,v \in V$:

$(1): \quad p - p = 0$
$(2): \quad p + 0 = p$
$(3): \quad p + u = p + v \iff u = v$
$(4): \quad q - p = r - p \iff q = r$


Proof

  • $(1): \quad p - p = 0$:

We have:

\(\displaystyle \left({p - p}\right) + \left({q - p}\right)\) \(=\) \(\displaystyle \left({p + \left({q - p}\right)}\right) - p\)
\(\displaystyle \) \(=\) \(\displaystyle q - p\)

From Zero Element is Unique:

$p - p = 0$


  • $(2): \quad p + 0 = p$

Using $(1)$ we see that:

\(\displaystyle p + 0\) \(=\) \(\displaystyle p + \left({p - p}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle p\)


  • $(3): \quad p + u = p + v \iff u = v$

Let $u = v$.

By definition a mapping has a unique image point on a given element.

It follows that:

$p + u = p + v$

Let $p + u = p + v$.

We have:

\(\displaystyle p + u\) \(=\) \(\displaystyle p + v\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left({p + u}\right) - p\) \(=\) \(\displaystyle \left({p + v}\right) - p\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left({p - p}\right) + u\) \(=\) \(\displaystyle \left({p - p}\right) + v\)
\(\displaystyle \implies \ \ \) \(\displaystyle u\) \(=\) \(\displaystyle v\) by $(1)$


  • $(4): \quad q - p = r - p \iff q = r$:

Let $q = r$.

By definition a mapping has a unique image point on a given element.

It follows that:

$q - p = r - p$

Let $q - p = r - p \in V$.

Then

\(\displaystyle q - p\) \(=\) \(\displaystyle r - p\)
\(\displaystyle \implies \ \ \) \(\displaystyle p + \left({q - p}\right)\) \(=\) \(\displaystyle p + \left({r - p}\right)\)
\(\displaystyle \implies \ \ \) \(\displaystyle q\) \(=\) \(\displaystyle r\) By the hypothesis $q - p = r - p$

$\blacksquare$