Understanding Transitivity of a Set: An Example

[evinda]

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)

 

[Sudharaka]

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 $y\in A$.

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.

 

[Evgeny.Makarov]

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.

 

[evinda]

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...

 

[Evgeny.Makarov]

Ah, I see that this is an old thread. I thought that you asked this question again recently.

 

[evinda]

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)

 

[Sudharaka]

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. :)