Proof involving a closed set of integers

[annoymage]

Homework Statement

proove is either true of false

let A be a set of integer closed under subtraction. if x and y are element of A, then x-ny is in A for any n in Z.

Homework Equations

n/a

The Attempt at a Solution

is there any proof, without induction?

i suspect its true because any arbitrary positive integer n will satisfy,

though if i try using induction also i stuck.

when n=0, satisfied,

assume it is true for some n>=0
x-(n+1)y=(x-ny)-y, clearly it is inside A

let n>0

x-(-n)y and i don't know how to continue now,

anyway, help me with this induction and also what are other ways without using proof by induction?

 

[Mark44]

Homework Statement

proove is either true of false

let A be a set of integer closed under subtraction. if x and y are element of A, them x-ny is in A for any n in Z.

Homework Equations

n/a

The Attempt at a Solution

is there any proof, without induction?

i suspect its true because any arbitrary positive integer n will satisfy,

though if i try using induction also i stuck.

when n=0, satisfied,

assume it is true for some n>=0
x-(n+1)y=(x-ny)-y, clearly it is inside A

let n>0

x-(-n)y and i don't know how to continue now,

anyway, help me with this induction and also what are other ways without using proof by induction?

If x and y are in A, then x - y is in A, by assumption that A is closed under subtraction.

Base case: n = 2
Let y be in A. We know that x - y is in A, so since A is closed under subtraction, then (x - y) - y is in A, and x - y - y = x - 2y.

Induction hypothesis: n = k
Assume the statement is true for n = k. I.e., assume that x - ky is in A

Now show that x - (k + 1)y is in A, using the induction hypothesis.

 

[annoymage]

x-(k+1)y = x-ky-y, then x-ky is in A(from induction hypothesis), y is in A,

so (y-ky)-y is also in A(closed under subtraction) ,

conclusion, x-ny is in A, for all integer n greater or equal to 2

but how to show the negative integer of n?

let n>0

x-(-n)y and i don't know how to continue now,

 

[Dick]

x-(k+1)y = x-ky-y, then x-ky is in A(from induction hypothesis), y is in A,

so (y-ky)-y is also in A(closed under subtraction) ,

conclusion, x-ny is in A, for all integer n greater or equal to 2

but how to show the negative integer of n?

let n>0

x-(-n)y and i don't know how to continue now,

Try something! Is 0 in A?

 

[annoymage]

Try something! Is 0 in A?

if A is closed under subtraction, let x be in A, then x-x=0 must be in A, so 0 is in A. is that correct?
but i don't have idea why this is related to induction, T_T

 

[Dick]

if A is closed under subtraction, let x be in A, then x-x=0 must be in A, so 0 is in A. is that correct?
but i don't have idea why this is related to induction, T_T

That's correct. Good. It's not related to the induction. It's related to solving your problem of why x-n*y being true for positive n is also true for negative n. x-(-n)y=x-n(-y). Now keep thinking.

 

[annoymage]

hmm, maybe , but i realized, set of integer that is closed under subtraction is either set containing 0, or set of integer itself,
ie; A = {0} or A = Z,
is that correct??

 

[annoymage]

huh, my i got counterexample for my statement, but i got some idea, let me think some more

 

[Dickfore]

Use mathematical induction.

 

[Dick]

Oh come on. You've already proved x-ny is in A using induction for n>=0. You want to prove x-(-n)y is in A for n>=0. x-(-n)y=x-n(-y)=x-n(0-y). I'm going to have to ask you to think again. If I have to get cute this is going to get ugly. I heard that in Futurama last night. Please see it. And by the way, the set of even numbers is closed under subtraction. So A isn't equal to either {0} or Z. I'm surprised you didn't see that either.

 

[annoymage]

aha, you sound like you have high expectation on me.. but much misunderstanding here, which we don't bother to know, anyway
if x and y is in A then -y in A (zero is in A), then x-n(-y) is in A (closed under subtraction),

owho, i get it already, thank you very much^^

 

[Dick]

aha, you sound like you have high expectation on me.. but much misunderstanding here, which we don't bother to know, anyway



if x and y is in A then -y in A (zero is in A), then x-n(-y) is in A (closed under subtraction),

owho, i get it already, thank you very much^^

There you go. You should also notice you can use that argument to show 'closed under subtraction' implies that it's also closed under addition.