Joining finite sequences

Question

How do I describe the joining of two finite sequences in mathematical notation? For example, suppose the following:

$$ A=(a_i)_{i=1,2}=(4,2)\\ B=(b_i)_{i=1,2}=(9,5)\\ C=(c_i)_{i=1,...,4}=(4,2,9,5) $$

Sequence $C$ can be considered sequence $A$ with sequence $B$ attached to the end. How do I describe sequence $C$ in terms of sequence $A$ and $B$?

$C= A \cup B$. There is no "attaching to the end". Sets are not ordered. — mjb, Jun 18 '13 at 13:07
@mjb I'm a bit confused. Why wouldn't a sequence be considered ordered? — William Breathitt Gray, Jun 18 '13 at 13:10
@VilhelmGray See my answer below. Sequences are ordered, sets are not. Your notation suggests you mean sets. — mjb, Jun 18 '13 at 13:11

score 2 · Accepted Answer · answered Jun 18 '13 at 13:10

2

Those are sets, not sequences. If you ment sets, then it is $C = A \cup B$, the union of the two sets.

If you ment finite sequences you could write \begin{equation} c_i = \begin{cases} a_i & i=1,2 \\ b_{i-2} & i=3,4 \end{cases} \end{equation}

answered Jun 18 '13 at 13:10

mjb

I'm sorry, I must have inadvertently used the notation for sets; I do mean sequences however. I want to correct the notation in my question, but I'm not sure what the correct notation for sequences are -- do I just remove the curly braces? – William Breathitt Gray Jun 18 '13 at 13:12
1

Then the correct notation would be: $A=(a_i){i=1,2}=(4,2)$, $B=(b_i){i=1,2}=(9,5)$ and $C=(c_i)_{i=1,\ldots,4}$ with $c_i$ defined as above. – mjb Jun 18 '13 at 13:15
There are of course more complicated notations by interpreting those sequences as functions $\lbrace 1,2 \rbrace \to \mathbb{R}$ or as aborting elements of $\ell^\infty$. – mjb Jun 18 '13 at 13:25

1 Answers1