Question

Which sets of numbers are closed under subtraction?

Choose all answers that are correct.

A.
odd natural numbers

B.
rational numbers

C.
{0, 1}

D.
{0, 1, 2}

Answer 1

@KonaFirestar @wio @dan815

Answer 2

That's a fun one. Do you recall the definition of what it means to be closed under an operation?

Answer 3

no

Answer 4

So, to be "closed" basically means that when you perform the operation on the numbers in your set, your answer stays inside the set. Let me give an example.

Answer 5

{0, 1, 2} is NOT closed under addition for example, because while 0 + 1 = 1 is in the set, 0 + 2 = 2 is in the set, and 1 + 1 = 2 is in the set, 2 + 1 = 3 is NOT in the set.

Answer 6

Does that sort of make sense of what we mean by closed?

Answer 7

yes

Answer 8

Alright, so let's try your question with subtraction

Answer 9

Start with C and D, those will be easier to rule in or out :)

Answer 10

So, is {0, 1} closed under subtraction?

Answer 11

no

Answer 12

Good, why not?

Answer 13

because 1 is not in the set

Answer 14

you mean -1 (negative 1) right? Because 0 - 1 = -1 not in the set.

Answer 15

yes sry :p

Answer 16

no probs :) just checking

Answer 17

OK, cool, how about D?

Answer 18

yes i think

Answer 19

So, {0, 1, 2} is NOT closed... for the same reason 0 - 1 = -1 not in the set (still!) :)

Answer 20

Make sense so far? Ready for A and B?

Answer 21

yes

Answer 22

Alright, so are the odd natural numbers closed under subtraction?

Answer 23

In other words, is the set {1, 3, 5, 7, 9, ...} closed under subtraction?

Answer 24

"natural" means positive... just FYI :)