I'm a beginner in set theory and I have doubt regarding mathematical induction. I came across the following examples.
Example 1:
Find the set given by the following definition:
1) $ 3 \in P $
2) For $x,y \in P, x + y \in P $
3) Only those elements obtained from steps (1) and (2) are in $P$
Solution: The set $P$ consists of positive integers which are multiples of 3.
Example 2:
Give an inductive definition of the set
$ P = \{2,3,4,\ldots\} = N - \{0,1\}$
Solution:
1) $ 2 \in P$ and $3 \in P $
2) If $x,y \in P$, then $x+y \in P$.
3) Only those elements obtained from steps 1 and 2 are in $P$.
Could anyone explain the above two examples and how to write this kind of definitions? I googled it, but couldn't get what I need. Thanks in advance.