Bridge to Abstract Mathematics: Mathematical Proof and Structures

6.3 EXISTENCE AND UNIQUENESS (OPTIONAL) 219

The distinction between the types of existence proof given in Examples

9 and 10 (existence of an object proved, but no rule provided in the proof

telling us how to calculate the object in particular cases) and those in Exam-

ples 7 and 8 is the basis of a famous controversy in mathematics that peaked

in the early part of this century, but has recently reemerged. In brief, an

existence proof in which we produce the object in question explicitly (we

omit a detailed discussion of exactly what this means) is said to be a con-

structive proof. A school of mathematicians known as intuitionists, founded

by the Dutch mathematician L. E. J. Brouwer (1 88 1 - l966), promulgated

the belief that the only allowable proofs of existence should be constructive

proofs. In particular, proofs by contradiction and proofs relying on such

axioms as the well-ordering principle (cf., Examples 9 and 10) should not

be regarded as legitimate. The intuitionist point of view failed to gain the

influence held by the formalist school, led by the great German mathema-

tician David Hilbert (1862-1943), whose program provided the framework

in which most of modern mathematics has developed. The debate still

rages however, as you can discover for yourself by consulting the January

1985 issue of the College Mathematics Journal. In that issue, a forum led

by Dr. Stephen B. Maurer and entitled "The algorithmic way of life is best"

provides ample evidence not only that mathematics is not a "closed book,"

but also that it is a developing field about whose major directions reason-

able and well-informed people can disagree strongly.

Exercises

1. (a) Prove that there exists a unique negative real number x satisfying the
   equation 44- = 2.
   (b) Prove that there does not exist a real number x satisfying the equation
   JO+J0=4.
   *(c) Prove that the equation = x + 8 has a unique real solution.
   (d) Prove that the equation 4- = x + 5 has a unique real solution.


2. (a) Recall from Example 3 the definition of a complement of a set. Prove that
   every set has at least one complement. Conclude from Example 3 that every set
   has a unique complement.
   (b) Given sets A and B from a universal set U, a subset C of U is called a com-
   plement of A relative to B (or a relative complement of A in B) if and only if A u C =
   A u B and A n C = a. Prove that, given any sets A and B, A has a unique
   relative complement in B.


3. (a) It is an axiom (additive identity axiom) for the real number system that there
   exists a real number, denoted 0, having the property that x + 0 = 0 + x = x for
   all x E R. Prove that zero is the only real number y
   x + y = y + x = x for each real number x.
   (b) Another axiom for R (multiplicative identity axiom)
   real number, denoted 1, such that x. 1 = 1. x = x for
   the only real number having this property.

having the property that

asserts the existence of a

all x E R. Prove that 1 is