MHB Transitive Sets: Prove, Show With $n$ Elements

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A transitive set is defined as one where all elements are subsets of the set itself. It is proven that if \( A \) is a transitive set, then \( A \cup \{A\} \) is also transitive because the only new member added, \( A \), is a subset of the original set. To demonstrate the existence of transitive sets with \( n \) elements, induction is used, starting with \( A_1 = \{\emptyset\} \) and defining \( A_{n+1} = A_n \cup \{A_n\} \). This construction ensures that for every natural number \( n \), there is a corresponding transitive set. The discussion highlights the foundational properties of transitive sets and their construction through induction.
Also sprach Zarathustra
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Hello, I need a help with the following:

1. Let $A$ be a transitive set, prove that $A\cup \{A \}$ is also transitive.
2. Show that for every natural $n$ there is a transitive set with $n$ elements.
 
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Also sprach Zarathustra said:
Hello, I need a help with the following:

1. Let $A$ be a transitive set, prove that $A\cup \{A \}$ is also transitive.
2. Show that for every natural $n$ there is a transitive set with $n$ elements.
For 2., use induction. Let $A_1 = \{\emptyset\}$. For $n\geqslant1$, let $A_{n+1} = A_n\cup \{A_n\}$ and use 1.
 
A transitive set is one in which all elements are subsets, now for 1. you have that the only new member that you have introduced is $A$ and it is a subset so the set is transtitve.

Imagine the tansitive set to be $A=\{1,2,3,4,5\}$ where these are defined in the usual way (in terms of the empty set).

Then the new set would be $B=\{1,2,3,4,5,A\}$ now then we can see that $A\in B$ but also that $\{1,2,3,4,5\}\subset B$ and so $A$ is a subset of B and so the set is transitive
 

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