Tuples

Ordered Pairs

Definition: Ordered Pair

The ordered pair $(a;b)$ of two objects $a$ and $b$ is the collection

$$(a;b)\stackrel{\text{def}}{=}\{\{a\},\{a,b\}\}$$

Theorem: Equality of Ordered Pairs

Two ordered pairs $(a;b)$ and $(c;d)$ are equal if and only if $a=b$ and $c=d$.

$$(a;b)=(c;d)⟺(a=b\wedge c=d)$$

PROOF

TODO

NOTE

This property of ordered pairs means that, in general, $(a;b)\ne (b;a)$, hence the name “ordered”.

Tuples

Definition: $n$-Tuple

An $n$-tuple $(a_1,\cdots ,a_n)$ is a collection of $n$ ordered pairs $\{(1,a_1),\cdots ,(n,a_n)\}$.

Theorem: Equality of $n$-Tuples

Two $n$-tuples $(a_1,\cdots ,a_n)$ and $(b_1,\cdots ,b_n)$ are equal if and only if $a_1=b_1,\cdots ,a_n=b_n$:

$$(a_1,\cdots ,a_n)=(b_1,\cdots ,b_n)⟺a_1=b_1,\cdots ,a_n=b_n$$

PROOF

TODO