Question

State whether each of the following sets have closure for + - x and division. (Natural #s), (Rational #s) (irrational #s) (whole #s)

Answer 1

do you know what "closure" means?

Answer 2

Yes

Answer 3

ok then what about \(\mathbb{N}\)
if you add two natural numbers, do you get a natural number?

Answer 4

Yes.

Answer 5

k then it is closed under \(+\)
if you subtract one natural number from another, do you always get a natural number?

Answer 6

I sorta need these quick... I have 3 more pages of homework left o_O

Answer 7

then lets do them quick
if you subtract, do you get a natural number?

Answer 8

Yes

Answer 9

what about \(5-10\)?

Answer 10

Oh... I see what u did there. lol Then it's not closed.

Answer 11

right

Answer 12

Multi is Closed, right

Answer 13

if you multiply two natural numbers, do you get a natural number?
yes, it is closed under \(\times\)

Answer 14

And division can't be a negative

Answer 15

division
is \(2\div 7\) a natural number?

Answer 16

Nope... so it's not closed

Answer 17

k
now since you want to be quick, rationals are closed under addition, subtraction
\[\frac{a}{b}\pm\frac{c}{d}=\frac{ad\pm bc}{bd}\]

Answer 18

Thank you for this!!

Answer 19

also multiplication
\[\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bc}\]

Answer 20

yw

Answer 21

division is a bit trickier
it is closed where the operation is defined, so i would say yes

Answer 22

in fact the answer is "yes" closed under division

Answer 23

as for irrationals, that is trickier still

Answer 24

you might think they were closed under addition, but they are not
for example
\[2+\sqrt{3}\] is irrational , but so is \(-\sqrt3\) and if you add them you get \(2\)

Answer 25

so irrationals not closed under addition, and of course therefore not closed under subtraction
as for multiplication, it is clear that they are not closed
take \(\sqrt{2}\times \sqrt{2}\) and get \(2\)

Answer 26

likewise for division

Answer 27

Thank you so much! I do have a couple other problems similar to this, they have various numbers in brackets. Not sure how to solve them. I will write a few

Answer 28

as for "whole numbers" i have never in my too long life understood the distinction between "natural" and "whole"
there is some argument about what is what, and i stay out of it

Answer 29

go ahead and post
if i know the answer i will help

Answer 30

{1} {0,1} {-1,0,1} {0,2,4,6....} {1,2,3} {1,3,5} {-1,1} that's all of them, each one with the brackets is a new problem and not part of the old one. Same directions at the top, I'm supposed to State whether the following sets have closure in each 4 ways

Answer 31

ok \(\{1\}\) is pretty clearly not closed under addition since \(1+1=2\)
damn sometime math is not so hard

Answer 32

neither under subtraction since \(1-1=0\)
however, it is closed under multiplication and division right?
\[1\times 1=1,\frac{1}{1}=1\]

Answer 33

you got the hang of this?

Answer 34

Yep, it's just I don't understand with multiple numbers in the bracket what I'm supposed to do. Plus I have a ton more homework due tomorrow morning

Answer 35

\(\{-1,0,1\}\) not closed under addition right? since \(1+1=2\)

Answer 36

or subtraction
but if you multiply any two of those numbers you get another one
also if you divide (with the understanding that you cannot divide by 0)

Answer 37

Ok great so multiply and divide are closed

Answer 38

even whole numbers
add any two you get another even whole number
subtract, no because you get \(2-2=-2\)

Answer 39

multiply, yes, the product of two evens is even
divide, no \(\frac{4}{4}=1\)

Answer 40

{1,2,3}
i don't think this is closed under anything
you can check though

Answer 41

likewise for {1,3,5}

Answer 42

as for {-1,1} i think it is closed wrt multiplication and division

Answer 43

Thank you!