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

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  • #1
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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. :/
 

Answers and Replies

  • #2
Dick
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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. :/
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?
 
  • #3
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A set containing multiples of 3 would be the other choice?
 
  • #4
Dick
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  • #5
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: > Thanks!
 
  • #6
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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.
 
  • #7
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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.
 
  • #8
Dick
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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.
 
  • #9
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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
Dick
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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
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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
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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
Dick
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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
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So, T={0,4 , 3 , 6, 9, ...}?
 
  • #15
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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
Dick
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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
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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
Dick
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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
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Thank you all.
 
  • #20
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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
Dick
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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
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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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