Express alternation of two sets

Question

Let's say I have two sets $X=\{1,2,3\}$ and $Y = \{A,B,C\}$, which do not have to be finite, and I want to express an infinite alternating series of these two sets like (example with no pattern):

$$ s = 2,B,3,A,1,C,2,C... $$

How would I express the domain of a series like this? I'm struggling with modeling the alternation. If I write something like $s \in (X \cup Y)^\omega$ I don't have the alternation at all. In $s \in (X \times Y)^\omega$ I deal with pairs which I don't want either. So what do you think is the best way to write down the domain of such a series?

For your set $B$, check out https://math.stackexchange.com/questions/502259/why-cannot-a-set-be-its-own-element — Václav Mordvinov, Mar 04 '19 at 09:03

score 1 · Accepted Answer · answered Mar 04 '19 at 09:32

1

The domain of an infinite series is N.
For each odd n, pick some element from X and
for each even n, pick some element from Y.

s in { f:N -> X $\cup$ Y : for all n in N, f(2n - 1) in X, f(2n) in Y }
where N is the positive integers.

answered Mar 04 '19 at 09:32

William Elliot

18,025

Thank you for this. I came up with the same idea but I thought there might be a more compact way to write that down. – Cilenco Mar 05 '19 at 13:42

Express alternation of two sets

1 Answers1