Hello, I need a help with the following:

1. Let $A$ be a transitive set, prove that $A∪{A}$ is also transitive.

2. Show that for every natural $n$ there is a transitive set with $n$ elements.

Thank you all!

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- Oct 27th 2012, 12:52 PMAlso sprach ZarathustraTransitive sets.
Hello, I need a help with the following:

1. Let $A$ be a transitive set, prove that $A∪{A}$ is also transitive.

2. Show that for every natural $n$ there is a transitive set with $n$ elements.

Thank you all! - Oct 27th 2012, 01:07 PMHallsofIvyRe: Transitive sets.
The term "transitive set" refers to sets of sets: The set, A, is said to be transitive if and only if whenever $\displaystyle x\in A$ and $\displaystyle y\in x$ then $\displaystyle y\in A$. (Frankly, I had to look that up!)

Now, if $\displaystyle x\in A\cup\{A\}$, then either $\displaystyle x\in A$ or x= A. Let $\displaystyle y\in x$ and consider those two cases.

As for 2, recall Von Neumann's definition of the natural numbers: 0 is the empty set, 1 is the set whose only member is the 0, 2 is the set whose only members are 0 and 1, etc.