Find Particular Set, 0 and n+3 belong to it and are N

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Homework Help Overview

The problem involves finding a set T of natural numbers that includes 0 and satisfies the condition that if n is in T, then n + 3 must also be in T. Additionally, T must not be equal to another set S, which is defined as the smallest set containing 0 and satisfying the same properties.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the nature of set T and its relationship to set S, with some suggesting that T could be all natural numbers or a specific subset like multiples of 3. Others question the implications of including certain numbers in T and how that affects the conditions set forth in the problem.

Discussion Status

The discussion is ongoing, with participants exploring various interpretations of the sets involved. Some have proposed examples of T that meet the criteria, while others are clarifying the definitions and constraints of the sets. There is no explicit consensus, but productive dialogue is occurring around the properties of T and S.

Contextual Notes

There is some confusion regarding the definition of natural numbers in this context, particularly whether 0 is included. Participants are also examining the implications of including specific numbers in T and how that aligns with the requirements of the problem.

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Homework Statement


Find a set T of natural numbers such that 0 ∈ T, and whenever n ∈ T,
then n + 3 ∈ T, but T ≠ S, where S is the set defined:
Define the set S to be the smallest set contained in N and satisfying the following two properties:
1. 0 ∈ S, and
2. if n ∈ S, then n + 3 ∈ S.

Homework Equations


I can only think that this set T is not the smallest set contained in N and satisfying the properties above.

The Attempt at a Solution


I have no clue how to find this. I can only say that T is not S, because it is given.
Please help. :/
 
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knowLittle said:

Homework Statement


Find a set T of natural numbers such that 0 ∈ T, and whenever n ∈ T,
then n + 3 ∈ T, but T ≠ S, where S is the set defined:
Define the set S to be the smallest set contained in N and satisfying the following two properties:
1. 0 ∈ S, and
2. if n ∈ S, then n + 3 ∈ S.

Homework Equations


I can only think that this set T is not the smallest set contained in N and satisfying the properties above.

The Attempt at a Solution


I have no clue how to find this. I can only say that T is not S, because it is given.
Please help. :/

T is any set satisfying those properties. The simplest one is ALL natural numbers. There is a smaller one that also satisfies it. Can't you think what that might be?
 
A set containing multiples of 3 would be the other choice?
 
knowLittle said:
A set containing multiples of 3?

Yes!
 
: > Thanks!
 
Dick said:
Yes!

S is the set of all multiples of 3.

T also has to contain all multiples of 3, but has to be unequal to S, so it should contain at least one number that's not a multiple of 3.
 
willem2 said:
S is the set of all multiples of 3.

T also has to contain all multiples of 3, but has to be unequal to S, so it should contain at least one number that's not a multiple of 3.

You are right. So, S could be ={1, 3, 6, 9, ...}
Thanks again.
 
knowLittle said:
You are right. So, S could be ={1, 3, 6, 9, ...}
Thanks again.

No, it can't. The definition says 0 has to be in S. And if 1 is in S then 1+3 has to be in S. Etc.
 
Dick said:
No, it can't. The definition says 0 has to be in S. And if 1 is in S then 1+3 has to be in S. Etc.

But, then why do you say the T can be all N numbers?
 
  • #10
knowLittle said:
But, then why do you say the T can be all N numbers?

I assume you are defining the natural numbers to include 0. Otherwise the whole problem makes no sense.
 
  • #11
I am confused. Even if 0 is included, then we could have that T is all N. Because the constraint that n+3 also belongs to S is there. So, T is not all natural numbers?
 
  • #12
knowLittle said:
Define the set S to be the smallest set contained in N and satisfying the following two properties:
1. 0 ∈ S, and
2. if n ∈ S, then n + 3 ∈ S.

The set has to have 0 in it. N as defined above has 0 in it.
Pick any natural number n. Then n + 3 will also be a natural number. For example, if you picked 1, then 1 + 3 = 4, which is also a natural number. If you picked, say 7, then 7 + 3 = 10 is also a natural number.

The problem, though, is that N is not the smallest set that works.
 
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  • #13
knowLittle said:
I am confused. Even if 0 is included, then we could have that T is all N. Because the constraint that n+3 also belongs to S is there. So, T is not all natural numbers?

There are many sets satisfying the definition of T. There is only one set satisfying the definition of S. S={0,3,6,9,...}. Just name an example of T that isn't equal to S.
 
  • #14
So, T={0,4 , 3 , 6, 9, ...}?
 
  • #15
knowLittle said:
So, T={0,4 , 3 , 6, 9, ...}?

Nope. Just go back to the requirements and check carefully and you will see why it can't be T.
 
  • #16
knowLittle said:
So, T={0,4 , 3 , 6, 9, ...}?

If 4 is in T then 4+3 must be in T. Read the definition of T. That doesn't work. You need to add a lot more numbers.
 
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  • #17
Ok. I understand that 4 alone cannot be in T.
A valid T set can be T={0, 3, 4, 6, 7, 9, 10, 12, 13,...}
I think I got it now.
 
  • #18
knowLittle said:
Ok. I understand that 4 alone cannot be in T.
A valid T set can be T={0, 3, 4, 6, 7, 9, 10, 12, 13,...}
I think I got it now.

Exactly. I think you've got it now too.
 
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  • #19
Thank you all.
 
  • #20
Dick said:
T is any set satisfying those properties. The simplest one is ALL natural numbers.
Thinking back, I have one more doubt.
Why do ALL N satisfy?

It would be all ## \forall \mathbb{N} \geq 3 ##. Right?
 
  • #21
knowLittle said:
Thinking back, I have one more doubt.
Why do ALL N satisfy?

It would be all ## \forall \mathbb{N} \geq 3 ##. Right?

Which one of the properties of T do you think ALL N doesn't satisfy??
 
  • #22
My professor said that as the problem is stated it does not contain all N, but we could add them. Nevermind, I think that he just said that without thinking. I see why one can plug in any N number in T, provided that we also insert its n-induction with n+3, (n+3)+3, and so on.

Thank you and sorry for the insecure question.
 

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