Bridge to Abstract Mathematics: Mathematical Proof and Structures

5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR +. 149

Figure 5.1 Triangles ABO and DCO, jkom

Example I, can be proved to be congruent.

It is generally through plane geometry that students are introduced to

mathematics as a deductive science. In that subject we begin with a set of

axioms, general statements about relationships in plane geometry that are

assumed true, and build from these a collection of theorems. Theorems are

deduced by means of a proof, a series of statements, each of whose validity

is based on an axiom or a previously proved theorem. In constructing

plane geometry proofs, we gain an appreciation of the critical importance

of the question, "What facts am I allowed to use in this proof?" This

question continues to be important in higher level college mathematics and

seldom, at that level, is the issue ever again as clearcut as it was in high

school geometry, since the latter is a self-contained and generally very

tightly constructed system. Often, when proofs are assigned in an upper-

level mathematics course, it is a fair question to ask of the instructor whether

a particular theorem or approach (perhaps from a previous course) may be

assumed and used in your proof.

EXAMPLE 2 (Intermediate Algebra) Use the associative and commutative

laws of multiplication to prove that (ab)(cd) = [(dc)a]b for any real num-

bers a, b, c, and d.

Solution Assume that a, b, c, and d are real numbers. We note that:

(ab)(cd) = (ab)(dc) (since cd = dc, by commutativity, and by the

basic principle, "equals multiplied by equals

yield equals")

= (dc)(ab) (again, by commutativity, applied to the real

numbers ab and dc)

= [(dc)a]b (by associativity, applied to the real numbers (dc),

a, and b)

Usually in proofs such as the one in Example 2 students are introduced
to a very basic, but crucial, idea: We cannot prove a general theorem (e.g.,
one involving universal quantijication over an in.nite universal set, such as
the real numbers) by giving a particular example, or by trying to enumerate
all cases. Note also the form of the proof given in Example 2. To prove