Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

Is there any formal notation for dealing with lists, rather than sets?

e.g. if I have a set $X=\{x_1,\dots,x_n\}$ and I want to add a new item to the set, say $x_{n+1}$, I can say "Let $X = X \cup \{x_{n+1}\}$" and it is clearly understood that I want to add $x_{n+1}$ to my set.

However, if $X$ is not a set but rather a list, or tuple (i.e. the elements are ordered and duplicates are allowed), is there any way of indicating that I am adding an element to the end of the list?

e.g. given $X=(x_1,\dots,x_n)$, how do I say add an element to $X$ such that $X=(x_1,\dots,x_n,x_{n+1})$? i.e. how do I formally denote appending an element to $X$?

share|cite|improve this question
When you say "Let $X=X\cup x_{n+1}$" you are (i) using programmer's lingo, not mathematical notation (mathematically, that only works if $x_{n+1}$ is a subset of $X$) and (ii) formally incorrect (you really want $X\cup\{x_{n+1}\}$, not $X\cup x_n$). – Arturo Magidin May 5 '11 at 17:13
To answer your actual question, we talk about "appending" $x_{n+1}$ to the tuple. – Arturo Magidin May 5 '11 at 17:13
Correct; actually I just messed up my latex and forgot to escape the brackets; i meant to write $X = X\cup \{x_{n+1}\}$. – TJ Ellis May 5 '11 at 17:31
Also, I purposefully didn't use the word "appending" because I thought that was programming lingo :-P – TJ Ellis May 5 '11 at 17:34

What you call a list is formally known as sequence. There was a question which symbol is for sequence concatenation. Unfortunately there is no accepted answer. Symbols , (commentator actually used u2322, "frown" symbol but it's resisting my attempt to copy it) and are mentioned in comments.

According the Wikipedia article is an operator for concatenation of numbers (doesn't specify which set of numbers, probably ℕ) but doesn't say much about sequences. The same symbol is in my opinion more commonly used for parallelism so it may confuse the reader.

I haven't seen symbol before but commentators agree about it.

share|cite|improve this answer

I don't think there is any standard notation.

One alternative would be to not use $(a,b)$ for ordered pairs but $a \times b$, which is the notation suggested by category theory. The $\times$ allows you to sweep lots of assocativity isomorphisms under the rug: it looks perfectly natural to write $(a \times b) \times c = a \times (b \times c) = a \times b \times c$, but not $((a,b),c) = (a,(b,c)) = (a,b,c)$.

Then if you have an $n$-tuple $x$ in $X^n$, you can write $x \times a$ for the $(n+1)$-tuple in $X^{n+1}$ obtained by appending $a$.

share|cite|improve this answer

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.