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!
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.