I am learning set theory right now and I am struggling to get to grips with definitions of sets involving predicates. For example, can someone tell me what "typical" elements look like in this set?
$$S = \left\{ {x^{n_0}yx^{n_1}\cdots yx^{n_k}\in A^n:k\ge0, P(k,n_0,n_1,\dots,n_k)}\right\}$$
Where $A^n$ is the set of, shall we say, "words" of length $n$ (the powers all add up to $n$ and let's not assume the $x$'s and $y$'s commute at all), and $P(k,n_0,n_1,\dots,n_k)$ I am happy to be anything to help me understand this definition. For example, what if we say that $P(k,n_0,\ldots,n_k)$ is satisfied when $n_0+\cdots+n_k$ is even?
\Now, on one hand, I when I think about the set $S$ I think about what elements look like when $k=0,1,2,3,\dots$ separately. Is this a good idea?
For example, when $k=0$, do all elements in $S$ have the form $x^{n_0}y$ or $x^{n_0}$ or both or maybe forms like these and also $x^{n_0}y^3$, etc? Does $y^n\in S$? What about $x^2y^{n-2}?$
Many thanks.
This is an exceptionally tricky bit of notation: in general things with an ellipsis like this should be more explicit (or at least not allow $k=0$). Thinking about $k=0,1,2,3,\ldots$ separately is a fine idea.
The definition of $S$ should also really specify some condition on the $n_i$. Are they positive? nonnegative? just integers? Perhaps the answer to that is hidden in the condition $P$ or the context. Since you say "polynomials", I'm going to assume nonnegative powers.
In any case, I think that since $k=1$ probably has things like $x^2 y x^4$ (if $n=7$), $k=0$ probably only allows $x^n$, with no $y$s at all. In general, $k$ should be the number of $y$s, given the notation you've written.
$y^n\in S$ as a $k=n$ element if my assumption on the $n_i$ was correct. $x^2y^{n-2}$ is a $k=n-2$ element if my assumption was correct. Something like $x^{n_0}y$ would be a $k=1$ element (and must have $n_0=n-1$), etc.