MHB Understanding Transitivity of a Set: An Example

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A set is defined as transitive if all elements of its elements are also members of the set. The set of natural numbers, denoted as $\omega$, is a classic example of a transitive set. Each natural number is also a transitive set, as it contains all smaller natural numbers. The discussion includes a debate about whether the set of rational numbers, $\mathcal{Q}$, is transitive, with a consensus that it is not since integers are subsets, not elements, of $\mathcal{Q}. The original poster successfully prepared for their set theory exam following this discussion.
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Hi! (Smile)

According to my notes, a set $A$ is called transitive if the elements of its elements are elements of $A$.
For example, the set of natural numbers $\omega$ is a transitive set.

Also, if $n \in \omega$ then $n$ is a transitive set since $n=\{0,1,2, \dots, n-1 \}$ and if we take a $k \in n$ then $k=\{0,1,2, \dots, k-1 \}$.

Could you give me an example of an other transitive set? (Thinking)
 
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evinda said:
Hi! (Smile)

According to my notes, a set $A$ is called transitive if the elements of its elements are elements of $A$.

Yes. So a set $A$ is transitive if whenever $x\in A$ and $y\in x$ then $y\in A$.

evinda said:
For example, the set of natural numbers $\omega$ is a transitive set.

Also, if $n \in \omega$ then $n$ is a transitive set since $n=\{0,1,2, \dots, n-1 \}$ and if we take a $k \in n$ then $k=\{0,1,2, \dots, k-1 \}$.

Could you give me an example of an other transitive set? (Thinking)

Another example would be the set of all rational numbers $\mathcal{Q}$. If you take any subset of the rational numbers like the integers for example, for each $y\in\mathcal{Z}$ we have $y\in \mathcal{Q}$ since any integer is a rational number.
 
Sudharaka said:
Another example would be the set of all rational numbers $\mathcal{Q}$. If you take any subset of the rational numbers like the integers for example, for each $y\in\mathcal{Z}$ we have $y\in \mathcal{Q}$ since any integer is a rational number.
I don't think $\Bbb Q$ is transitive. In your explanation, $\Bbb Z\subseteq\Bbb Q$ and not $\Bbb Z\in\Bbb Q$.

Evinda, I see that you have asked this question on Math.StackExchange. Why are you asking it again?

P.S. "Another" is one word.
 
Evgeny.Makarov said:
Evinda, I see that you have asked this question on Math.StackExchange. Why are you asking it again?

I had asked it before here in mathhelpboards but I didn't get an answer and since I wanted to know the answer since I would write a test in Set theory the day after , I asked it also in Math.StackExchange...
 
Ah, I see that this is an old thread. I thought that you asked this question again recently.
 
Evgeny.Makarov said:
Ah, I see that this is an old thread. I thought that you asked this question again recently.

(Smile) I did well in the exams in Set theory.. Thanks for your help! (Happy)
 
Evgeny.Makarov said:
I don't think $\Bbb Q$ is transitive. In your explanation, $\Bbb Z\subseteq\Bbb Q$ and not $\Bbb Z\in\Bbb Q$.

Evinda, I see that you have asked this question on Math.StackExchange. Why are you asking it again?

P.S. "Another" is one word.

Yep. My mistake. That doesn't work. Thanks for pointing that out. :)
 

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