$egingroup$ other is "closed under to fill in the blank" if applying fill in the empty to aspects of other yields aspects of something. $endgroup$
$egingroup$ 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

2,08511 yellow badge1414 silver- badges2323 bronze title
include a comment |
(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

2,3021515 silver badges2121 bronze title
add a comment |

your Answer

Thanks for contributing solution to stclairdrake.netematics Stack Exchange!

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

But avoid

Asking for help, clarification, or responding to other answers.Making statements based upon opinion; earlier them up with recommendations or personal experience.

Use stclairdrake.netJax to format equations. stclairdrake.netJax reference.

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

See more: Can Parents Kick You Out At 18 If Still In School Student? Am I Responsible For My Child After They Turn 18

Draft saved
Draft discarded

Sign up or log in in

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

Post as a guest

email Required, however never shown

Post as a guest


Required, but never shown

write-up Your price Discard

By clicking “Post her Answer”, you agree to our regards to service, privacy policy and also cookie policy

Upcoming events
Featured ~ above Meta
closeup of the door under limited union and also Closed under countable union
What walk the "closed over"/"closed under" terminology typical exactly and where did that come from?
What does "relatively closed" mean?
closeup of the door under countable union
Difference in between closed under pairwise intersection and closed under limited intersection.
What go "index sets are filtered" mean?
What go "closed under subsets" mean?
What does being closed under "extension that its members" mean?
warm Network concerns more hot questions

concern feed
i ordered it to RSS
question feed To subscribe to this RSS feed, copy and paste this URL right into your RSS reader.

ridge Exchange Network
site style / logo © 2021 ridge Exchange Inc; user contributions licensed under cc by-sa. Rev2021.11.11.40719

stclairdrake.netematics stack Exchange works best with JavaScript permitted

your privacy

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