Inverses for Real Addition
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Theorem
Each element $x$ of the set of real numbers $\R$ has an inverse element $-x$ under the operation of real number addition:
- $\forall x \in \R: \exists -x \in \R: x + \left({-x}\right) = 0 = \left({-x}\right) + x$
Proof
We have:
\(\displaystyle \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] + \left({-\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]}\right)\) | \(=\) | \(\displaystyle \left[\!\left[{\left \langle {x_n - x_n} \right \rangle}\right]\!\right]\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \left[\!\left[{\left \langle {0_n} \right \rangle}\right]\!\right]\) |
Similarly for $\left({-\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]}\right) + \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$.
Thus the inverse of $x \in \left({\R, +}\right)$ is $-x$.
$\blacksquare$