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

- anonymous

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- anonymous

do you know what "closure" means?

- anonymous

Yes

- anonymous

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

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## More answers

- anonymous

Yes.

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

Yes

- anonymous

what about \(5-10\)?

- anonymous

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

- anonymous

right

- anonymous

Multi is Closed, right

- anonymous

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

- anonymous

And division can't be a negative

- anonymous

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

- anonymous

Nope... so it's not closed

- anonymous

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

- anonymous

Thank you for this!!

- anonymous

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

- anonymous

yw

- anonymous

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

- anonymous

in fact the answer is "yes" closed under division

- anonymous

as for irrationals, that is trickier still

- anonymous

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 \(2\)

- anonymous

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 \(2\)

- anonymous

likewise for division

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

{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

- anonymous

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

- anonymous

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

- anonymous

you got the hang of this?

- anonymous

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

- anonymous

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

- anonymous

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)

- anonymous

Ok great so multiply and divide are closed

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

likewise for {1,3,5}

- anonymous

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

- anonymous

Thank you!

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