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?