I'm currently using the following notation to denote a sequence (i.e. ordered list of elements):
$\langle x_n | x \in \mathbb{N} \rangle$
E.g. $S = \langle 1,3,5,7 \rangle$ and $S_2 = 3$
I know other notations exist, such as $\{S_k\}_{k\in\mathbb{N}}$ or $(S_k)$, but this doesn't really affect my following question.
What is the correct way to construct a new sequence from an existing sequence with some elements removed? I used to denote this as follows:
$S' = S \setminus \{1,5\} = \langle 3,7 \rangle$
But the set difference operator is not well-defined for a sequence and a set of elements. Since a sequence is commonly defined as a function $f:\mathbb{N} \mapsto E$ with $E$ the domain (set of elements which can be contained in the sequence), I was wondering if there is a better way to denote what I want to express here — maybe there exists some operator I'm missing?
Thanks.
Best Answer
I don't think there's a truly general standard notation. For the removal of single elements, I have sometimes seen the following notation:
$$(s_0, \ldots, \widehat{s_i}, \ldots, s_n),$$ meaning that the $i$-th element was removed.
The most general way would be to use sub-sequences: Define a sequence $i: \mathbb N \to \mathbb N$ that maps to the indices you want to keep, and write $s_{i_j}$.