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

knowLittle

## 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.

Homework Helper

## 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.

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?

knowLittle
A set containing multiples of 3 would be the other choice?

Homework Helper
A set containing multiples of 3?

Yes!

knowLittle
: > Thanks!

willem2
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.

knowLittle
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.

Homework Helper
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.

knowLittle
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?

Homework Helper
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.

knowLittle
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?

Mentor
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.

1 person
Homework Helper
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.

knowLittle
So, T={0,4 , 3 , 6, 9, ...}?

Hornbein
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.

Homework Helper
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.

1 person
knowLittle
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.

Homework Helper
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.

1 person
knowLittle
Thank you all.

knowLittle
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?

Homework Helper
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??

knowLittle
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.