Tuple
Let \( I \) be an index set, and \( S \) a set then we say that a function \( t : I \to S \) is a tuple,
Tuple at an Index
Suppose that \( t \) is a tuple and \( i \in \operatorname{ dom } \left( t \right) \) , then we define that the tuple's value at index \( i \) as \( t \left( i \right) \)
Tuple Subscript Notation
Suppose that \( t \) is a tuple, then we define \( t _ i \) to be \( t \) at index \( i \)
Ordered Tuple Notation
Suppose that the index set has some natural ordering, then the notation \( \left( t _ { i _ 1 } , t _ { i _ 2 } , \ldots \right) \)
Finite Tuple
A finite tuple is a tuple with a finite index set
Naturally Indexed Tuple
A naturally indexed tuple is a tuple whose index set equals \( \mathbb{ N } _ 1 \) or \( [ 1 \ldots m ] \) for some \( m \in \mathbb{ N } _ 1 \)

Note that a naturally indexed tuple is represented using ordered tuple notation using \( \left( a _ 1, a _ 2, \ldots \right) \). When it's finite we get \( \left( a _ 1, a _ 2, a _ 3, a _ 4 \right) \)

Finite Tuple Equality
We define two \( n \) tuples \( (x_1, x_2, ..., x_n) \) and \( (y_1, y_1, ..., y_n) \) to be equal when \[ \forall i \in [1 \ldots n], x_i = y_i \]

Note that finite tuples tuples are indexed starting from 1

Length
Suppose that \( a = \left( a _ 1, a _ 2, \ldots , a _ n \right) \) then \( \operatorname{ len } \left( a \right) = n \)
Zero Indexed Tuple
A zero indexed tuple is a tuple of the form \( a = \left( a _ 0, a _ 1, \ldots a _ m \right) \)
Length of a Zero Indexed Tuple
Suppose that \( a = \left( a _ 0, a _ 1, \ldots a _ m \right) \) is a zero indexed tuple, then \( \operatorname{ len } \left( a \right) = m + 1 \)
Converting from Regular to Zero Indexing
Let \( a \) be a finite tuple and suppose \( \operatorname{ len } \left( a \right) = n \) , let \( r \) be \( a \) but indexed regularly, and let \( z \) be \( a \) but using zero indexing, then for all \( i \in [ 1 ... n ], r _ i = z _ { i - 1 } \)
Reverse of a Finite Tuple
Suppose that \(a = \left( a _ 1, a _ 2, \ldots , a _ n \right) \), is an \( n \) tuple, then we define \[ \operatorname{rev} \left( a \right) := \left( a _ n, \ldots , a _ 1 \right) \]
Reverse Cancels
For any \( n \) tuple \( a \) \[ \operatorname{ rev } \left( \operatorname{ rev } \left( a \right) \right) = a \]
Concatenation
Suppose that \( a = \left( a _ 1, \ldots , a _ n \right) \) and \( b = \left( b _ 1, \ldots , b _ m \right) \) are finite tuples, then \[ \operatorname{ concat } \left( a, b \right) = \left( a _ 1, \ldots a _ n, b _ 1, \ldots , b _ m \right) \]
Sorted in Ascending Order
Suppose that \(a = \left( a _ 1, a _ 2, \ldots , a _ n \right) \in \mathbb{ R } ^ n \), then we say that \( a \) is sorted in ascending order when for any \( i \in [ 1 \ldots n - 1 ] \) \( a _ i \le a _ { i + 1 } \)
Ascending Tuple
Show that \( \left( 1, 2, 3, 4, 100 \right) \) is an ascending \( n \) tuple
Sorted in Descending Order
Suppose that \(a = \left( a _ 1, a _ 2, \ldots , a _ n \right) \in \mathbb{ R } ^ n \), then we say that \( a \) is sorted in descending order when for any \( i \in [ 1 \ldots n - 1 ] \) \( a _ i \ge a _ { i + 1 } \)
Descending Tuple
Show that \( \left( 8, 5, 3, 1, 0 \right) \) is a descending \( n \) tuple
The reverse of Ascending is Descending
Suppose that \( a \) is a finite tuple sorted in ascending order, then \( \operatorname{ rev } \left( a \right) \) is sorted in descending order.
Inversion
Suppose that \( \left( a _ 1, a _ 2, \ldots , a _ n \right) \in \mathbb{ R } ^ n \), then an inversion is a tuple \( i , j \in [ 1 \ldots n ] \times [ 1 \ldots n ] \) such that \( i \lt j \) but \( a _ i \gt a _ j \)

Given the tuple \( a := \left( 8, 9, 10, 11, 3 \right) \) then we can see that \( \left( 1, 5 \right) \) is an inversion because \( 1 \lt 5 \) but \( 8 \gt 3 \)