$\begingroup$ we say miscellaneous is close up door under operation x if using operation x come a set of aspects y yields facets in y. $\endgroup$
SteveKass well perhaps unlimited sums need to be thought about a various operation altogether. Is such thing as using an procedure infinitely plenty of times well defined? $\endgroup$
A collection is closeup of the door under addition if you can add any two numbers in the set and still have actually a number in the set as a result. A collection is closed under (scalar) multiplication if you deserve to multiply any two elements, and also the result is still a number in the set.
For instance, the set $\1,-1 \$ is closed under multiplication however not addition.
You are watching: Which of the following sets is closed under division?
I typically see "closed under some operation" together the elements of the collection not gift able to "escape" the set using the operation.
Usually (not generally) it requires an operation, for example: the organic numbers room closed under addition way that if I include two organic numbers, the amount will additionally be a natural number. This same set is not closed under subtraction since $1-2=-1$, and $-1$ is no a herbal number
Usually the empty is filled v an "operation". For instance you have actually a set $S = \a,b,c,d,... \ $ i beg your pardon is close up door under some procedure $ \star $
Which means: $ \star : S \times S \to S $ or in words: You may pick any two facets of $S$, apply $ \star$ top top them and also they can be assigned a new value in $S$. So come say: You are 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 set $S$ and division $\star$ as your operation.
Now you have : $4 \star 2 = 2 \in S$, i beg your pardon is fine. However you also have: $4 \star 3 \notin S$ together $4 \star 3$ together by our definition would be the portion $\frac43$
Most common operations space addition, multiplication etc. For the herbal numbers, integers, genuine numbers etc.. Yet you don"t need to be so details and can define your collection 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 concern has good answers already, yet I perform not see the price that ns expected, so ns am writing this.)
I great to include a official definition. Let $X$ be a set, $n\in\couchsurfingcook.combbN$ (BTW, $0\in\couchsurfingcook.combbN$). $f$ is an $n$-ary operation on $X$ iff $f$ is a function from $X^n$ to $X$. Let $Y$ be a subset of $X$. $Y$ is closed under $f$ iff for every $a\in Y^n$ $f(a)\in Y$.
Remarks. As you see, a closed collection ($Y$ in this definition) is a subset the another set ($X$ in this definition), and the operation might take and also give members that $X$ which room not in $Y$. Every set $Z$ is closeup of the door under every $n$-ary operation on $Z$, therefore 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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