$egingroup$ we say miscellaneous is close up door under procedure x if applying operation x come a set of elements y yields aspects in y. $endgroup$
SteveKass fine perhaps boundless sums should be considered a various operation altogether. Is such thing as applying an procedure infinitely plenty of times fine defined? $endgroup$
A set is closed under addition if friend can add any two numbers in the collection and still have a number in the collection as a result. A collection is close up door under (scalar) multiplication if you deserve to multiply any two elements, and also the an outcome is tho a number in the set.
For instance, the collection $1,-1 $ is close up door under multiplication but not addition.
You are watching: Is the set of integers closed under addition
I normally see "closed under some operation" as the aspects of the set not gift able to "escape" the collection using that operation.
Usually (not generally) it entails an operation, because that example: the natural numbers space closed under addition method that if I include two herbal numbers, the amount will likewise be a natural number. This same set is no closed under subtraction because $1-2=-1$, and also $-1$ is not a organic number
Usually the empty is filled v an "operation". For example you have actually a set $S = a,b,c,d,... $ i beg your pardon is closeup of the door under some procedure $ star $
Which means: $ star : S imes S o S $ or in words: You may pick any two aspects of $S$, apply $ star$ on them and they deserve to be assigned a brand-new value in $S$. So to say: You room not leaving your set $S$ by utilizing this operation.
However, in general, this go not have to be the case: You may pick the integers as your set $S$ and department $star$ as your operation.
Now you have actually : $4 star 2 = 2 in S$, which is fine. But you likewise have: $4 star 3 otin S$ together $4 star 3$ as by our an interpretation would be the portion $frac43$
Most common operations space addition, multiplication etc. Because that the organic numbers, integers, real numbers etc.. However you don"t have to be so particular and can specify your collection and your operation arbitrarily.
edited Mar 3 "19 at 19:03
answer Mar 1 "16 at 19:38
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(This inquiry has great answers already, however I perform not watch the answer that ns expected, so ns am composing this.)
I wish to add a official definition. Permit $X$ it is in a set, $ninaramuseum.orgbbN$ (BTW, $0inaramuseum.orgbbN$). $f$ is an $n$-ary operation on $X$ iff $f$ is a role from $X^n$ come $X$. Permit $Y$ be a subset the $X$. $Y$ is closed under $f$ iff for every $ain Y^n$ $f(a)in Y$.
Remarks. As you see, a closed set ($Y$ in this definition) is a subset of another collection ($X$ in this definition), and also the operation might take and give members of $X$ which room not in $Y$. Every collection $Z$ is close up door under every $n$-ary operation on $Z$, so the hatchet “closed under” is useless as soon as $Y=X$.
reply Dec 31 "17 at 17:30
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