Equality of Ordered Tuples/Examples/Ordered Triple
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Example of Equality of Ordered Tuples
Let:
- $\tuple {a_1, a_2, a_3}$ and $\tuple {b_1, b_2, b_3}$
be ordered triples.
Then:
- $\tuple {a_1, a_2, a_3} = \tuple {b_1, b_2, b_3}$
- $\forall i \in \set {1, 2, 3}: a_i = b_i$
Proof 1
A special case of Equality of Ordered Tuples for $m = n = 3$.
$\blacksquare$
Proof 2
\(\ds A\) | \(=\) | \(\ds B\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \tuple {a_1, a_2, a_3}\) | \(=\) | \(\ds \tuple {b_1, b_2, b_3}\) | Definition of $A$ and $B$ | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \tuple {a_1, \tuple {a_2, a_3} }\) | \(=\) | \(\ds \tuple {b_1, \tuple {b_2, b_3} }\) | Definition of Ordered Triple | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a_1\) | \(=\) | \(\ds b_1\) | Equality of Ordered Pairs | ||||||||||
\(\, \ds \land \, \) | \(\ds \tuple {a_2, a_3}\) | \(=\) | \(\ds \tuple {b_2, b_3}\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a_1\) | \(=\) | \(\ds b_1\) | Equality of Ordered Pairs | ||||||||||
\(\, \ds \land \, \) | \(\ds a_2\) | \(=\) | \(\ds b_2\) | |||||||||||
\(\, \ds \land \, \) | \(\ds a_3\) | \(=\) | \(\ds b_3\) |
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product sets: Exercise $4$
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions: Exercise $5$