(1) A set S of integers is defined recursively by the rules: (1) 1 ? S, and (2) If n ? S, then 2n + 1 ? S. (a) Is 20 ? S? (b) Is 175 ? S? Explain your answers. (2) A set, S, of strings over the alphabet ? = {a,b,c} is defined recursively by (1) a ? S and (2) if x ? S then xbc ? S. List all the strings in S of length seven or less. 8 (3) A set, S, of positive integers is defined recursively by the rule: (1) 3 ? S, and (2) If n ? S, then 2n ? 3 ? S. List all the elements in the set S. (4) Give a recursive definition of the set of positive integers that end with the digit 7. (5) Describe the strings in the set S of strings over the alphabet ? = {a,b,c} defined recursively by (1) c ? S and (2) if x ? S then xa ? S and xb ? S and cx ? S. nt: Your description should be a sentence that provides an easy test to check if a given string is in the set or not.
An example of such a description is: S consists of all strings of aÂ’s, bÂ’s, and cÂ’s, with more aÂ’s than bÂ’s. That isnÂ’t a correct description since cab is in S and doesnÂ’t have more aÂ’s than bÂ’s, and also baac isnÂ’t in S, but does have more aÂ’s than bÂ’s. So that attempted description is really terrible. The best way to do this problem is to use the rules to build a bunch of strings in S until a suitable description becomes obvious. (6) (bonus) A set S of ordered pairs of integers is defined recursively by (1) (1,2) ? S and (2,1) ? S, and (2) if (m,n) ? S, then (m+2,n) ? S, and (m,n+2) ? S, and (m+1,n+1) ? S. There is a simple description of the ordered pairs in S. What is it? Your description should be good enough so that you can instantly decide if (12236,912242) is in S. Reach me for the correct answer, price is negotiable 20usd-45usd