Unit 14 Answers

1. { 1, 2, 3, ..., n }

2.   A_n

3.     Basis Clause: 5 S.

       Inductive Clause: If   x, y S,  then   x + y S

       Extremal Clause: Nothing is in S unless it is obtained from the Basis and Inductive Clauses.

4.

1.        Basis Clause: 0 S.

          Inductive Clause: If   x S,  then  x + 2 S   and   x - 2 S

       Extremal Clause: Nothing is in S unless it is obtained from the Basis and Inductive Clauses.

2.        Basis Clause: 2 S.

          Inductive Clause: If  x S,  then   x + 3 S

       Extremal Clause: Nothing is in S unless it is obtained from the Basis and Inductive Clauses.

3.        Basis Clause: 1 S,   2 S,   3 S,   4 S,  

          Inductive Clause: If   x S,  then   x + 5 S

          Extremal Clause: Nothing is in S unless it is obtained from the Basis and Inductive Clauses.

5. When the string consists of   n 0's   followed by   n 1's   for some nonnegative integer n.