Lists

For a non negative integer $n$, a list $x$ of length $n$ is an ordered collection of $n$ elements (which can repeat) denoted by $(x_{1}, x_{2}, \ldots, x_{n}$. Two lists are equal if and only if they have the same number of elements and in the same order.\newline

Lists are different from sets because firstly the order of elements matters, and secondly, the elements can repeat in a list.\newline

We denote an empty list as $()$ which has length $0$ and is also a trivial case of some of the theorems later.