Bridge to Abstract Mathematics: Mathematical Proof and Structures

148 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5

to, or can quickly be grasped by, students at the sophomore level of under-
graduate mathematics.

Conclusions Involving V, but Not 3 or +. Proof by Transitivity

Proof by Transitivity

In this article we focus on proofs of the most elementary type from a log-
ical, or structural, point of view. Generally, theorems whose conclusion
involves neither the existential quantifier nor implication arrow are proved
by methods familiar to students with a strong high school mathematics
background. Since these methods continue to be useful at every level of
mathematics, their proper application in the context of sophomore-junior
level university mathematics is an appropriate starting point for our study
of theorem-proving techniques.
Most students of mathematics are first exposed to proof writing in the
plane geometry, intermediate algebra, and trigonometry courses that pre-
cede the introduction to calculus. Here are typical proofs from each of the
three courses.

EXAMPLE 1 (Plane Geometry) Hypothesis: BC and AD are straight lines,
AB = DC, 0 bisects BC, angle B = 90°, angle C = 90". Conclusion: A0 =
DO. Plan: Prove that A0 and DO are corresponding parts of congruent
triangles. See Figure 5.1.

Solution

Statements

In A ABO and ADCO;

AB = BC

0 bisects BC

Therefore BO = CO

Angle B = 90°, angle C = 90"

Therefore angle B = angle C

Therefore A ABO is

congruent to A DCO

A0 and DO are corresponding

sides of triangles A ABO and

A DCO

Therefore A0 = DO

Authorities

By hypothesis

By hypothesis

Definition of bisector

By hypothesis

Axiom "two quantities

equal to the same quantity

are equal to each other."

SAS (two sides, included

angle)

A0 and DO lie opposite

equal angles.

Corresponding sides of

congruent triangles are

equal.