$\begingroup$ miscellaneous is "closed under to fill in the blank" if using fill in the blank to aspects of something yields facets of something. $\endgroup$
$\begingroup$ 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.

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

2,09611 yellow badge1414 silver- badges2323 bronze badges
include a comment |
(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 $Y=X$.

point out
answered Dec 31 "17 at 17:30

2,2721515 silver- badges2121 bronze title
add a comment |

your Answer

Thanks because that contributing an answer to couchsurfingcook.comematics ridge Exchange!

Please be certain to answer the question. Carry out details and share her research!

But avoid

Asking for help, clarification, or responding to various other answers.Making statements based on opinion; earlier them increase with recommendations or an individual experience.

Use couchsurfingcook.comJax to layout equations. Couchsurfingcook.comJax reference.

To learn more, watch our tips on writing an excellent answers.

See more: Save The Last Dance Full Movie 123Movies, Save The Last Dance

Draft saved
Draft discarded

Sign increase or log in

authorize up utilizing Google
sign up utilizing Facebook
sign up utilizing Email and Password

Post together a guest

email Required, but never shown

Post as a guest


Required, but never shown

article Your price Discard

By click “Post your Answer”, you agree come our regards to service, privacy policy and cookie plan

The Overflow Blog
Featured ~ above Meta
closeup of the door under limited union and Closed under countable union
What does the "closed over"/"closed under" terminology median exactly and where did that come from?
What walk "relatively closed" mean?
close up door under countable union
Difference in between closed under pairwise intersection and also closed under limited intersection.
What go "index sets room filtered" mean?
What walk "closed under subsets" mean?
What does gift closed under "extension of its members" mean?
warm Network concerns more hot inquiries

inquiry feed
subscribe to RSS
question feed To subscribe to this RSS feed, copy and paste this URL right into your RSS reader.

ridge Exchange Network
site design / logo © 2021 ridge Exchange Inc; user contributions licensed under cc by-sa. Rev2021.10.1.40358

couchsurfingcook.comematics ridge Exchange works ideal with JavaScript permitted

your privacy

By click “Accept all cookies”, you agree stack Exchange deserve to store cookie on your machine and disclose information in accordance v our Cookie Policy.