Unit 14 Exercises

1. Let A_i = { 1, 2, 3, ..., i } for i = 1, 2, 3, ... . Find

    A_i

2. Let A_i = { i, i+1, i+2, ... } for i = 1, 2, 3, ... . Find

    A_i

3. Give a recursive definition of the set of positive integers that are multiples of 5.

4. Give a recursive definition of

1. the set of even integers.
2. the set of positive integers congruent to 2 modulo 3.
3. the set of positive integers not divisible by 5.

5. When does a string belong to the set A of bit strings (i.e. strings of 0's and 1's) defined recursively by

        Basis Clause: A

        Inductive Clause: 0x1 A   if x A

        where is the empty string (An empty string is a string with no symbols in it.)

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

Answers for these exercises