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

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No need to apologize! It's always good to ask questions and clarify any doubts. And yes, you are absolutely correct. T can contain any natural number, as long as we also include its n-induction with n+3. Keep up the good work!
  • #1
knowLittle
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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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  • #2
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?
 
  • #3
A set containing multiples of 3 would be the other choice?
 
  • #4
knowLittle said:
A set containing multiples of 3?

Yes!
 
  • #5
: > Thanks!
 
  • #6
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.
 
  • #7
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.
 
  • #8
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.
 
  • #9
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.
 

1. What is the meaning of "Find Particular Set, 0 and n+3 belong to it and are N"?

The phrase "Find Particular Set, 0 and n+3 belong to it and are N" is a mathematical statement that is asking to find a set of numbers where both 0 and n+3 are elements of the set and n is a natural number (N). This set can be represented as {0, n+3} where n is any natural number.

2. How can we determine the possible values of n in the given set?

The possible values of n can be determined by solving the equation n+3 = N, where N is any natural number. This equation can be rewritten as n = N-3, which means that n can take on any value that is 3 less than a natural number.

3. Is it possible for the set to have more than two elements?

Yes, it is possible for the set to have more than two elements. The only requirement is that both 0 and n+3 must be elements of the set. This means that the set can have additional elements as long as these two elements are included.

4. Can the set contain negative numbers?

Yes, the set can contain negative numbers. As long as both 0 and n+3 are elements of the set, the set can include any other numbers, including negative numbers. For example, the set {0, -1, 2} satisfies the given conditions where n = -1.

5. How can this concept be applied in real-world situations?

This concept can be applied in various real-world situations that involve sets and their elements. For example, in a store that sells items for $3 each, a customer has $0 to spend and needs to buy a certain number of items (represented by n) plus 3 more items to get a discount. Here, the set {0, n+3} represents the possible combinations of items that the customer can buy to get the discount.

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