$\begingroup$ miscellaneous is "closed under to fill in the blank" if using fill in the empty to aspects of miscellaneous yields elements of something. $\endgroup$
$\begingroup$ we say something is closeup of the door under operation x if using operation x come a set of aspects y yields aspects in y. $\endgroup$
SteveKass fine perhaps boundless sums have to be taken into consideration a different operation altogether. Is such thing as applying an operation infinitely many times well defined? $\endgroup$
A set is close up door under addition if girlfriend can include any two numbers in the set and still have actually a number in the set as a result. A set is close up door under (scalar) multiplication if you can multiply any kind of two elements, and the an outcome is quiet a number in the set.

For instance, the collection $\1,-1 \$ is close up door under multiplication yet not addition.

You are watching: What does it mean to be closed under addition

I normally see "closed under some operation" together the aspects of the set not being able to "escape" the collection using that operation. Usually (not generally) it entails an operation, for example: the natural numbers space closed under addition means that if I add two natural numbers, the amount will additionally be a natural number. This same collection is no closed under subtraction since $1-2=-1$, and also $-1$ is no a herbal number  Usually the empty is filled through an "operation". For instance you have a collection $S = \a,b,c,d,... \$ i m sorry is closeup of the door under some procedure $\star$

Which means: $\star : S \times S \to S$ or in words: You may pick any type of two facets of $S$, use $\star$ ~ above them and also they can be assigned a brand-new value in $S$. So come say: You space not leaving your collection $S$ by using 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 actually : $4 \star 2 = 2 \in S$, which is fine. However you likewise have: $4 \star 3 \notin S$ as $4 \star 3$ together by our definition would be the portion $\frac43$

Most usual operations are addition, multiplication etc. Because that the organic numbers, integers, real numbers etc.. However you don"t need to be so particular and can specify your set and your procedure arbitrarily.

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edited Mar 3 "19 in ~ 19:03
answered Mar 1 "16 at 19:38 ImagoImago
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(This question has good answers already, but I execute not watch the answer that ns expected, so ns am creating this.)

I wish to add a officially definition. Let $X$ it is in a set, $n\in\bsci-ch.orgbbN$ (BTW, $0\in\bsci-ch.orgbbN$). $f$ is one $n$-ary operation on $X$ iff $f$ is a function from $X^n$ to $X$. Permit $Y$ it is in a subset the $X$. $Y$ is closed under $f$ iff for every $a\in Y^n$ $f(a)\in Y$.

Remarks. Together you see, a closed set ($Y$ in this definition) is a subset of another set ($X$ in this definition), and also the operation may take and give members of $X$ which are not in $Y$. Every collection $Z$ is close up door under every $n$-ary operation on $Z$, for this reason the ax “closed under” is useless as soon as $Y=X$.

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answer Dec 31 "17 at 17:30 beroalberoal
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