Equality of Ordered Tuples/Examples/Ordered Triple

From ProofWiki
Jump to navigation Jump to search

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:

$\tuple {a_1, a_2, a_3} = \tuple {b_1, b_2, b_3}$

if and only if:

$\forall i \in \set {1, 2, 3}: a_i = b_i$


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