Integer Addition is Associative/Proof 2
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Theorem
The operation of addition on the set of integers $\Z$ is associative:
- $\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$
Proof
Let $a, b, c, d, e, f \in \N$ such that:
- $x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$.
Then:
\(\ds x + \paren {y + z}\) | \(=\) | \(\ds \eqclass {a, b} {} + \paren {\eqclass {c, d} {} + \eqclass {e, f} {} }\) | Definition of Integer | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {a, b} {} + \eqclass {c + e, d + f} {}\) | Definition of Integer Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {a + \paren {c + e}, b + \paren {d + f} } {}\) | Definition of Integer Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {\paren {a + c} + e, \paren {b + d} + f} {}\) | Natural Number Addition is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {a + c, b + d} {} + \eqclass {e, f} {}\) | Definition of Integer Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\eqclass {a, b} {} + \eqclass {c, d} {} } + \eqclass {e, f} {}\) | Definition of Integer Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x + y} + z\) | Definition of Integer |
$\blacksquare$