Question

A set of numbers is said to be closed under an operation if the result of combining any two numbers in the set results in a number that is also in the set. Decide whether or not each set is closed under the operation.
a. {positive integers]; division
b. {odd integets}; x
c. {odd integers}; +
d. {integers ending in 4 or 6}; x

Answer 1

Positive integers divided by positive integers will only give you positive integers

Answer 2

positive divided by positive will result in positive
positive divided by negative will result in negative
negative divided by negative willl result in a negative

Answer 3

For a. lemme give you an example: \(\Large\rm \dfrac{5}{3}\)
There is a positive integer, dividing a positive integer.
Is the result a positive `integer`?

Answer 4

Yeah.

Answer 5

No, 5/3 is not an integer.

Answer 6

That's a ratio, right?

Answer 7

Lemme pick some better numbers ^^ hehe
How bout 10 divided by 4?

`positive integer` divided by `positive integer`,
but the result is `2.5`, which is NOT an integer, right? :o

Answer 8

Yeah...

Answer 9

We have `left` the set of positive integers by using division.
So we're not closed under division for the first set, yah? :)

Answer 10

Oh yah.
So for b, the set is closed, right?

Answer 11

For b. and c. you need to remember a little something about even and odd numbers.
`odd` times `odd` = ?
`odd` plus `odd` = ?

3 x 7 = 21
5 x 5 = 25
9 x 3 = 27

`odd` times `odd` = `odd`
Yes, good :)

Answer 12

odd plus odd = positive

1+1=2
3+3=6

so is C, not closed?

Answer 13

Correct.
Because {odd integer} + {odd integer} =/= {odd integer}
Open.

Answer 14

Is D closed too?

Answer 15

lol are you just guessing? :)
How do you conclude that for D?

Answer 16

Haha, kind of :3

4x4=16
6x6=36
4x6=24

Answer 17

Ah good good.
You really only need to check 3 cases, yah?

I'm gonna use a sloppy notation a sec.
So if you have a number ending in 4, ...4
and multiply it by a number ending in 4,
...4 x ...4 = .....16

It ends in a 6, regardless of what the other numbers are.
That works out for the other cases as well.
Yay good job \c:/

Answer 18

^_^ Thank youu [: