I'm reading up on set theory and relation and I need help understanding this: Two sequences of the same element type can be composed to form a single sequence in such a way that the order of each sequence is maintained, and the elements of one sequence follow the elements of the other. The composition operator, representing concatenation, is defined as
Can someone explain this in simple English. I don't understand the notations.
I'm going to call the map $c$, not whatever that symbol is.
The first two lines mean $c$ is a function which takes a pair of sequences $s = (s_1, \dotsc, s_n)$ and $t = (t_1, \dotsc, t_m)$, where $n = \#s$ and $m = \#t$ are the lengths, and outputs a new sequence $c(s, t)$. The next lines tell us the elements of the sequence $c(s, t)$. This suffices as a definition because sequences are determined by their elements. If $1 \le i \le n$, the $i$th element of $c(s,t)$ is $s_i$. Any number bigger than $n$ can be written as $i + n$ for some $i$, so if $1\le i \le m$, the $(i+n)$th element of $c(s,t)$ is $t_i$. Personally, I would write this part as "if $n < i \le n + m$, the $i$th element of $c(s,t)$ is $t_{n-i}$".
What really is going on here is best illustrated by an example: $c((s_1, s_2, s_3), (t_1, t_2)) = (s_1, s_2, s_3, t_1, t_2)$.
All the remark about additivity is saying is that if you stick two sequences together, the length of the result is the sum of the length of the parts.