$egingroup$ we say other 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 unlimited sums should be taken into consideration a various operation altogether. Is such point as using an procedure infinitely numerous times well defined? $endgroup$
A collection is close up door under addition if friend can include any two numbers in the collection and still have actually a number in the collection as a result. A collection is closeup of the door under (scalar) multiplication if you deserve to multiply any kind of two elements, and the result 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: Which set is closed under subtraction?
I normally see "closed under some operation" as the aspects of the collection not gift able to "escape" the set using that operation.
Usually (not generally) it entails an operation, because that example: the herbal numbers are closed under addition method that if I include two natural numbers, the sum will likewise be a natural number. This same set is not closed under subtraction since $1-2=-1$, and $-1$ is not a natural number
Usually the empty is filled with an "operation". For instance you have actually a collection $S = a,b,c,d,... $ i m sorry is closed under some operation $ star $
Which means: $ star : S imes S o S $ or in words: You may pick any kind of two facets of $S$, use $ star$ top top them and they deserve to be assigned a new value in $S$. So come say: You space not leaving your collection $S$ by utilizing this operation.
However, in general, this does not need to be the case: You may pick the integers together your set $S$ and division $star$ as your operation.
Now you have actually : $4 star 2 = 2 in S$, i m sorry is fine. But you additionally have: $4 star 3 otin S$ as $4 star 3$ together by our meaning would be the portion $frac43$
Most typical operations are addition, multiplication etc. Because that the herbal numbers, integers, actual numbers etc.. However you don"t have to be so particular and can specify your collection and your procedure arbitrarily.
edited Mar 3 "19 in ~ 19:03
answer Mar 1 "16 at 19:38
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(This inquiry has an excellent answers already, however I carry out not view the prize that ns expected, so ns am writing this.)
I great to add a formal definition. Permit $X$ be a set, $ninstclairdrake.netbbN$ (BTW, $0instclairdrake.netbbN$). $f$ is one $n$-ary operation on $X$ iff $f$ is a duty from $X^n$ to $X$. Permit $Y$ be a subset the $X$. $Y$ 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 collection ($X$ in this definition), and also the operation might take and also give members of $X$ which space not in $Y$. Every set $Z$ is closed under every $n$-ary procedure on $Z$, for this reason the hatchet “closed under” is useless as soon as $Y=X$.
reply Dec 31 "17 at 17:30
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