**Unit 14 Exercises**

**1. **Let *A _{i}* = { 1, 2, 3, ..., i }
for

*A _{i}*

**2. **Let *A _{i} =* { i, i+1, i+2, ... }
for

*A _{i}*

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

**4.** Give a recursive definition of

- the set of even integers.
- the set of positive integers congruent to 2 modulo 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: 0*x*1
*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.