Here is a question about inductive defined functions and proofs. Please provide

Question

Here is a question about inductive defined functions and proofs. Please provide your complete solution, and thumbs up if your answer works.

note: the definition of base b representation:

How to prove inductively:

Thanks!

2. points N, where Y is (a) Give an inductive definition of the base b representation function 10, 1 11. (b) Prove that for every b 1, every natural number n has a base b interpretation. That is, inductively for all there a string z E X* such that (z)i, Identify where your proof uses the fact n, exists that b> 1.

Explanation / Answer

a) (.)b (wa) = (.)(w) * b+ a , where a belongs to Sigma

(.)b (a) = a, where a belongs to sigma.

b) Let the minimum no. which can not be written in base b is n then let's devide n by b we get n = bq+r where q and r are less than n (since b>1) so if q and r are representable in base b then we can represent n as well but n is not representable so one of q or r is also not representable hence nis not minimum. Hence contradicts our hypothesis hence done.

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