Question

Which sets of numbers are closed under addition?



Choose ALL answers that are correct.



A. whole numbers


B. natural numbers


C. negative integers


D. integers

Answer 1

do you know what closed means?

Answer 2

No..

Answer 3

@micahm

Answer 4

hello

Answer 5

Please help..

Answer 6

he natural numbers are well-ordered: which means every set of natural numbers has a least element.

so suppose S is a set of natural numbers closed under addition.

let k be the smallest element of S.

then S contains:

k,k+k, k+k+k,....etc

in other words S must contain all multiples of k.

could S contain other elements besides multiples of k?

suppose it did. suppose it contained m.

then we get all natural numbers of the form ak + bm.

for example, if k = 2, m = 3, S might be:

S = {2,3,4,5,6,7,8,.......} = N - {0,1}.

note we can write this set as:

{2 + k(gcd(2,3)): k in N}

this can be generalized to more than a pair of numbers....do you see how?

as you can see, we can come of with LOTS of sets this way.

Answer 7

ctually, there is an infinity of sets of natural numbers which are closed under addition.

Answer 8

So its natural numbers?

Answer 9

yes

Answer 10

But I need another answer.

Answer 11

Think about the set of all multiples of any given natural number n. E.g., the even natural numbers is the set of all multiples of 2.

What about, say, 17. Is the sum of two multiples of 17 a multiple of 17?

There are other, more complicated sets that are closed under addition. E.g., suppose a set contains 5 and 7. What's the smallest subset of N that is closed under addition that contains 5 and 7?

It's a bit more complicated to look at the set of natural numbers N. If you look at the integers, and are allowed to add and subtract, then the only sets closed under addition are multiples of a given number, and the set is related to the GCD of members

Answer 12

all of these sets are closed under addition

Answer 13

ALL of them?

Answer 14

if you add two of them you get another of them.
for example

add two negative numbers, the sum is negative. so negative numbers are closed

Answer 15

Okay...so whats the answer?

Answer 16

add two whole numbers, the sum is another whole number

Answer 17

add two integers, the sum is also an integer.

Answer 18

the natural numbers are not closed under subtraction operation. do you see why?

Answer 19

2 and 5 are natural numbers,
but 2 - 5 = -3 , and -3 is not a natural number

Answer 20

Okay..Yeah.

Answer 21

under addition many sets are closed

Answer 22

ok i have a question for you.
is the set
{ -1, 0,1} closed under addition?

Answer 23

@micahm sorry i didnt understand your solution, what do you mean by well ordered set, why do we need to use this to show closure . i assume youre proving this rigorusly

Answer 24

But I'm leaving for a trip..My plane is almost here I need the answer.

Answer 25

Lol okay

Answer 26

:/

Answer 27

One more please very quickly..

Answer 28

so what are the answers???????

Answer 29

Way to copy my comment.