\$egingroup\$ other is "closed under to fill in the blank" if using fill in the blank to facets of miscellaneous yields facets of something. \$endgroup\$
\$egingroup\$ us say something is closed under procedure x if using operation x come a collection of aspects y yields aspects in y. \$endgroup\$
SteveKass fine perhaps limitless sums need to be thought about a various operation altogether. Is such point as using an operation infinitely many times fine defined? \$endgroup\$
A collection is closeup of the door under addition if friend can add any two numbers in the set and still have actually a number in the set as a result. A collection is closeup of the door under (scalar) multiplication if you can multiply any type of two elements, and also the an outcome is quiet a number in the set.

For instance, the set \$1,-1 \$ is closeup of the door under multiplication however not addition.

You are watching: Is a set of real numbers closed under addition

I normally see "closed under some operation" together the aspects of the set not gift able to "escape" the set using that operation.

Usually (not generally) it requires an operation, for example: the natural numbers are closed under addition way that if I include two herbal numbers, the sum will likewise be a organic number. This same set is no closed under subtraction since \$1-2=-1\$, and \$-1\$ is not a organic number

Usually the blank is filled through an "operation". For instance you have a set \$S = a,b,c,d,... \$ which is closeup of the door under some procedure \$ star \$

Which means: \$ star : S imes S o S \$ or in words: You may pick any type of two elements of \$S\$, use \$ star\$ on them and also they have the right to be assigned a new value in \$S\$. So to say: You room not leaving your collection \$S\$ by making use of this operation.

However, in general, this walk not need to be the case: You might pick the integers as your collection \$S\$ and division \$star\$ together your operation.

Now you have : \$4 star 2 = 2 in S\$, which is fine. Yet you likewise have: \$4 star 3 otin S\$ as \$4 star 3\$ together by our definition would it is in the portion \$frac43\$

Most common operations room addition, multiplication etc. Because that the organic numbers, integers, real numbers etc.. Yet you don"t have to be so particular and can define your collection and your procedure arbitrarily.

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edited Mar 3 "19 in ~ 19:03
answer Mar 1 "16 in ~ 19:38

ImagoImago
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(This inquiry has great answers already, yet I execute not check out the answer that i expected, so i am composing this.)

I great to add a formal definition. Permit \$X\$ it is in a set, \$ninetterworld2016.orgbbN\$ (BTW, \$0inetterworld2016.orgbbN\$). \$f\$ is one \$n\$-ary operation on \$X\$ iff \$f\$ is a role from \$X^n\$ come \$X\$. Allow \$Y\$ be a subset of \$X\$. \$Y\$ is closed under \$f\$ iff because that every \$ain Y^n\$ \$f(a)in Y\$.

Remarks. As you see, a closed set (\$Y\$ in this definition) is a subset of another set (\$X\$ in this definition), and also the operation might take and also give members the \$X\$ which room not in \$Y\$. Every set \$Z\$ is closed under every \$n\$-ary operation on \$Z\$, so the hatchet “closed under” is useless as soon as \$Y=X\$.

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answered Dec 31 "17 at 17:30

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