The set of arithmetic expressions over the real numbers

Chapter 5, Problem 4E

(choose chapter or problem)

Problem 4E

The set of arithmetic expressions over the real numbers can be defined recursively as follows:

I. BASE: Each real number r is an arithmetic expression.

II. RECURSION: If u and v are arithmetic expressions, then the following are also arithmetic expressions:

a. (+u)

b. (−u)

c. (u + v)

d. (u − v)

e. (u ·v)

III. RESTRICTION: There are no arithmetic expressions over the real numbers other than those obtained from I and II.

(Note that the expression is legal even though the value of v may be 0.) Give derivations showing that each of the following is an arithmetic expression.