$\begingroup$ we say something is closeup of the door under operation x if using operation x come a set of aspects y yields aspects in y. $\endgroup$
SteveKass fine perhaps boundless sums have to be taken into consideration a different operation altogether. Is such thing as applying an operation infinitely many times well defined? $\endgroup$
A set is close up door under addition if girlfriend can include any two numbers in the set and still have actually a number in the set as a result. A set is close up door under (scalar) multiplication if you can multiply any kind of two elements, and the an outcome is quiet a number in the set.
For instance, the collection $\1,-1 \$ is close up door under multiplication yet not addition.
You are watching: What does it mean to be closed under addition
I normally see "closed under some operation" together the aspects of the set not being able to "escape" the collection using that operation.
Usually (not generally) it entails an operation, for example: the natural numbers space closed under addition means that if I add two natural numbers, the amount will additionally be a natural number. This same collection is no closed under subtraction since $1-2=-1$, and also $-1$ is no a herbal number
Usually the empty is filled through an "operation". For instance you have a collection $S = \a,b,c,d,... \ $ i m sorry is closeup of the door under some procedure $ \star $
Which means: $ \star : S \times S \to S $ or in words: You may pick any type of two facets of $S$, use $ \star$ ~ above them and also they can be assigned a brand-new value in $S$. So come say: You space 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 collection $S$ and division $\star$ together your operation.
Now you have actually : $4 \star 2 = 2 \in S$, which is fine. However you likewise have: $4 \star 3 \notin S$ as $4 \star 3$ together by our definition would be the portion $\frac43$
Most usual operations are addition, multiplication etc. Because that the organic numbers, integers, real numbers etc.. However you don"t need to be so particular and can specify your set and your procedure arbitrarily.
edited Mar 3 "19 in ~ 19:03
answered Mar 1 "16 at 19:38
2,09611 gold badge1414 silver- badges2323 bronze badges
add a comment |
(This question has good answers already, but I execute not watch the answer that ns expected, so ns am creating this.)
I wish to add a officially definition. Let $X$ it is in a set, $n\in\bsci-ch.orgbbN$ (BTW, $0\in\bsci-ch.orgbbN$). $f$ is one $n$-ary operation on $X$ iff $f$ is a function from $X^n$ to $X$. Permit $Y$ it is in a subset the $X$. $Y$ is closed under $f$ iff for every $a\in Y^n$ $f(a)\in Y$.
Remarks. Together you see, a closed set ($Y$ in this definition) is a subset of another set ($X$ in this definition), and also the operation may take and give members of $X$ which are not in $Y$. Every collection $Z$ is close up door under every $n$-ary operation on $Z$, for this reason the ax “closed under” is useless as soon as $Y=X$.
answer Dec 31 "17 at 17:30
2,2721515 silver- badges2121 bronze badges
add a comment |
Thanks because that contributing an answer to bsci-ch.orgematics ridge Exchange!Please be certain to answer the question. Administer details and also share her research!
But avoid …Asking for help, clarification, or responding to various other answers.Making statements based on opinion; earlier them up with references or personal experience.
Use bsci-ch.orgJax to style equations. Bsci-ch.orgJax reference.
To discover more, check out our advice on writing an excellent answers.
See more: How Many Cups In 1 Kilo Of Flour (Apf) Kg, How Many Kg Are In A Unit
Sign up or log in
sign up making use of Google
authorize up using Facebook
authorize up making use of Email and also Password
Post as a guest
email Required, but never shown
Post together a guest
Required, but never shown
article Your prize Discard
Featured ~ above Meta
closeup of the door under limited union and Closed under countable union
What does the "closed over"/"closed under" terminology typical exactly and also where did the come from?
What does "relatively closed" mean?
close up door under countable union
Difference between closed under pairwise intersection and also closed under limited intersection.
What does "index sets space filtered" mean?
What does "closed under subsets" mean?
What does gift closed under "extension the its members" mean?
hot Network concerns more hot inquiries
subscribe to RSS
concern feed To subscribe to this RSS feed, copy and also paste this URL right into your RSS reader.
stack Exchange Network
site style / logo design © 2021 ridge Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.9.24.40305