Unit 14 Exercises

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

    cup_n.gif 
(1023 bytes) Ai

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

    cap_n.gif (1019 bytes) Ai

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: Theta.gif (164 bytes) in.gif (889 bytes) A

        Inductive Clause: 0x1 in.gif (889 bytes) A   if x in.gif (889 bytes) A

        where Theta.gif (164 bytes) 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