# Transitive sets.

#### Also sprach Zarathustra

##### Member
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.

#### Opalg

##### MHB Oldtimer
Staff member
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.

#### hmmm16

##### Member
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