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July 28

Point wise union

If I have two sets of sets; {{1,2},{3}} and {{4},{5,6}} then the cartesian product is {({1,2},{4}),({1,2},{5,6}),({3},{4}),({3},{5,6})}. However I am after the set of sets that I get when I instead of producing ordered pairs take the union. So that I arrive at $\{\{1,2\}\cup \{4\},\{1,2\}\cup \{5,6\},\{3\}\cup \{4\},\{3\}\cup \{5,6\}\}=\{\{1,2,4\},\{1,2,5,6\},\{3,4\},\{3,5,6\}\}$ . Does this operation have a name? Taemyr (talk) 14:29, 28 July 2008 (UTC)[reply]

This may not be what you're looking for, but you can write the operation compactly as a set comprehension:

\{x\cup y:x\in A,y\in B\}

. -- BenRG (talk) 17:49, 28 July 2008 (UTC)[reply]

That set is the range (or image) of the union of those sets. Is that any help? --_{tiny plastic} Grey Knight ⊖ 20:36, 28 July 2008 (UTC)[reply]

This operation is useful in the study of set-theoretic ideals. It can be notated by a union sign with an underbar. I just call it pointwise union. --Trovatore (talk) 20:47, 28 July 2008 (UTC)[reply]

Then I'l go with that. Thx all. Taemyr (talk) 00:00, 29 July 2008 (UTC)[reply]

Notation for multivariate polynomials of at most degree m

I’m looking for a notation for the set of polynomials over a field $F$ in variables $x_{1},x_{2},...,x_{n}$ of total degree at most $m$ . Anyone know any? GromXXVII (talk) 19:41, 28 July 2008 (UTC)[reply]

It tends to be used for PDEs and mixed partial derivatives, but the Multi-index notation will serve. RayAYang (talk) 20:35, 28 July 2008 (UTC)[reply]

F_{m}[x_{1},x_{2},...,x_{n}]

or

F^{m}[x_{1},x_{2},...,x_{n}]

should work, as long as you define it the first you use it. The only issue I can see is that it isn't clear if the notation means degree at most m, or degree exactly m. And, of course, the second choice could be mistaken for meaning polynomials over a vector space, but that would be rather weird. --Tango (talk) 21:38, 28 July 2008 (UTC)[reply]