Equality of Ordered Triples/Proof 2
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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
\(\ds A\) | \(=\) | \(\ds B\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \tuple {a_1, a_2, a_3}\) | \(=\) | \(\ds \tuple {b_1, b_2, b_3}\) | 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
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$: Theorem $1.9$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory