# Set composed of operations on a Subset

Let there be a set P, and a set K such that $P\subset K$. Let there be 2 binary operations closed on K written $+$ and $\times$. Is there any way to define K as having only elements composed of applications of these operations on elements of P?

In other words, let $P = \{A,B,C\}$. K would then be defined as the set $K =\{A+A,A\times B,A+B\times C,\cdots\}$ and all other possible combinations of the operations on P.

## 1 Answer

The set $K$ is typically called the closure of $P$ under $+,\times$. We can define it recursively by setting $K_0=P$ and, given $K_n$, letting $K_{n+1}$ be $K_n\cup\{a+b\mid a,b\in K_n\}\cup\{a\times b\mid a,b\in K_n\}$. Finally, $K=\bigcup_n K_n$.

The same idea applies to any number of operations, regardless of whether they are binary or not. Usually, given a set $S$ closed under the operations, we would define the set $K$ you are interested in as the closure of $P$ inside $S$. And even more generality is possible, leading for instance to the idea of Skolem hulls in first order logic.

• Thank you. Is there any notation to write "K is the closure of P"? – Disousa Apr 14 '14 at 20:11
• Some people may write $\mathrm{Cl}(P)$, but I would recommend, if you decide to use the notation, to define it first, something along the lines of: "Let $\mathrm{Cl}(P)$ denote the closure of $P$ under the operations $+,\times$, that is, letting $K_0=P$ and, given $K_n$, setting $K_{n+1}=\dots$, then $\mathrm{Cl}(P)=\bigcup_n K_n$. This is the smallest set $T$ that contains $P$ and such that for any $p,q\in T$, we also have $p+1,p\times q\in T$." – Andrés E. Caicedo Apr 14 '14 at 20:16
• Rosser used $\mathrm{Clos}(P,r)$ for the closure of $P$ under a relation $r$. It can get a little clunky, but I think it's nicely unambiguous. Still a good idea to explain it when first introducing its use. – Malice Vidrine Apr 14 '14 at 20:53