Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


share|cite|improve this question
up vote 1 down vote accepted

If elements of $S$ are unique, then $S \setminus \{\ldots\}$ would be clear for me. If you define a sequence as $f : \mathbb{N} \to E$, then probably the easiest way to formally state your operation is as

$$ (f \ominus g)(k) = M(0,0,k)$$ where $$ M(a,b,k) = \begin{cases} M(a+1,b+1,k) &\text{if }f(a) = g(b) \\ M(a+1,b,k-1) &\text{if }f(a) \neq g(b) \land k > 0 \\ f(a) &\text{otherwise} \end{cases} $$

Note that here the you don't need $f$ to be injective, however, it is the left-most subsequence that matches $g$ that is removed.

Still, personally I would describe it using words for the sake of the reader. It is very rare that you need such a symbolic approach and keeping formalism to the necessary minimum often makes your text more approachable.

Hope that helps ;-)

share|cite|improve this answer
(The elements of $S$ are not necessarily unique.) Thanks -- I've made a few modifications, but your definition of $M(a,b,k)$ put me on the right track! – Macuyiko Mar 1 '13 at 14:15

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 $s_{i_j}$.

share|cite|improve this answer
Thanks for the help -- I'll keep the hat notation in mind. Alas, it is more than a single element which has to be removed. – Macuyiko Mar 1 '13 at 14:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.