Inverse 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:
- $\forall x \in \R: \exists -x \in \R: x + \paren {-x} = 0 = \paren {-x} + x$
Proof
We have:
\(\ds \eqclass {\sequence {x_n} } {} + \paren {-\eqclass {\sequence {x_n} } {} }\) | \(=\) | \(\ds \eqclass {\sequence {x_n - x_n} } {}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {\sequence {0_n} } {}\) |
Similarly for $\paren {-\eqclass {\sequence {x_n} } {} } + \eqclass {\sequence {x_n} } {}$.
Thus the inverse of $x \in \struct {\R, +}$ is $-x$.
$\blacksquare$