1. Recursive Definition

Hi All, new to this forum and need some help with a question on recursive definitions as this practice question has really got me baffled.

Base Case: 4 ∈ X, 6 ∈ X
Recursive Case: If x ∈ X and y ∈ X then (x-y) ∈ X.

State wether -4 ∈ X, 9 ∈ X and 10 ∈ X.

By my rubbish knowledge we would first do the following? (4-6) ∈ 2

So -2 ∈ X?

I'm not sure what we would do next? 6 - 8?

Any help would be great.

2. Re: Recursive Definition

hi richtea9

let's look at the first one: -4 ∈ X:

if you subtract: 6-4,you get 2 and from the recursive case we get that 2 ∈ X.

now subtract: 2-6,and you get -4 and again from the recursive case you get that -4 ∈ X.

the third case can be done similarly,maybe just with more subtractions.

but,for the second case you have to state that there is no way to get an odd number by subtracting two even numbers.to understand this try doing applying this to prove that 3 ∈ X is not true.insted of three you can use any odd number.

3. Re: Recursive Definition

Originally Posted by anonimnystefy
hi richtea9

let's look at the first one: -4 ∈ X:

if you subtract: 6-4,you get 2 and from the recursive case we get that 2 ∈ X.

now subtract: 2-6,and you get -4 and again from the recursive case you get that -4 ∈ X.

the third case can be done similarly,maybe just with more subtractions.

but,for the second case you have to state that there is no way to get an odd number by subtracting two even numbers.to understand this try doing applying this to prove that 3 ∈ X is not true.insted of three you can use any odd number.
Hi Anonimnystefy,

Thanks for the reply, I really appreciate it.

Am I right by thinking -4 ∈ X is true, but the other two 9 ∈ X and 10 ∈ X are false as the number is going to be continuously be negative after the first result (2)?

4. Re: Recursive Definition

hi

as i said the first is true and the second one is false,but look at this:

4-6=-2 ∈ X

6-(-2)=6+2=8 ∈ X

do you now see that the numbers aren't going to be only negative.you can subtract positive minus negative to get a greater positive number.

5. Re: Recursive Definition

Originally Posted by anonimnystefy
hi

as i said the first is true and the second one is false,but look at this:

4-6=-2 ∈ X

6-(-2)=6+2=8 ∈ X

do you now see that the numbers aren't going to be only negative.you can subtract positive minus negative to get a greater positive number.
Thanks again for the quick reply.

Sorry for my lack of my knowledge but are the following possible? (trying to get my head round this)

6 - (-2) = 8
8 - (-2) = 10
10 - (-2) = 12

-2 - 6 = -8
-8 - 6 = -14
-14 - 6 = - 20

6. Re: Recursive Definition

hi

your welcome!Feel free to hit the Thanks button !

yes,all of those are correct,and all of those are elements of X.what can you deduce from the numbers you got?

7. Re: Recursive Definition

Originally Posted by anonimnystefy
hi

your welcome!Feel free to hit the Thanks button !

yes,all of those are correct,and all of those are elements of X.what can you deduce from the numbers you got?
Thanks given.

That there all even?

8. Re: Recursive Definition

hi

yes they are all even,so that's how you conclude that the second prob. given to you (9 ∈ X) is false.