I started learning induction and was wondering if I'm getting it thus far.
An example is given, with 2 questions based on it.
Unary (Peano) numbers over the alphabet { 0, n, (, )} are inductively defined as follows:
1) 0 is an element of the collection,
2) If n is an element of the collection, then so is S(n).
Now I'm asked to give an alphabet and inductive definition of the decimal natural numbers and binary numbers.
Decimal:
Alphabet: { 0, 1, n, + }
1) 0 is an element of the decimal natural numbers,
2) If n is an element of the decimal natural numbers, then so is n + 1.
Binary:
Alphabet: { 1, 0, n }
1) 0 is an element of the binary numbers
2) 1 is an element of the binary numbers
3) If n is an element of the binary numbers ,then so is 0n
4) If n is an element of the binary numbers, then so is 1n
Is my method and choosing of alphabet correct?