Principles of Mathematics in Operations Research

6 1 Introduction

X

y

Z x

Fig. 1.2. Proposition 1.3.

Proof. Backward:

B: x = y (a; - y = 0) <=> FXZ = .XTZ (triangle is equilateral)

Forward:

A-(i) Area: ^a;j/ = ^-

A-(ii) Pythagorean Theorem: x^2 + y^2 = z^2

<£• \xy = ^±^ <£> a;^2 - 2xj/ + y^2 = 0 «• (a: - y)^2 = 0 <=> a; - y = 0. •

The above proof is a good example of how forward-backward combination

can be used. There are special cases defined by the forms of A or B with the

use of quantifiers. The first three out of four cases are based on conditions on

statement B and the last one arises when A has a special form.

Construction (3)

If there is an object (3a; € N) with a certain property(a: > 2) such that

something happens (x^2 — 5x + 6 = 0), this is a construction. Our objective

here is to first construct the object so that it possesses the certain property

and then to show that something happens.

Selection (V)

If something (3a; E I 3 2* = j) happens for every object (Vj/ € R+) with
a certain property (y > 0), this is a selection. Our objective here is to first
make a list (set) of all objects in which something happens (T — {y € M+ :
3a; e R 3 2X — y}) and show that this set is equivalent to the set whose
elements has the property (S = R+). In order to show an equivalence of two
sets (S — T), one usually has to show (S C T) and (T C S) by choosing a
generic element in one set and proving that it is in the other set, and vice
versa.

Specialization

If A is of the form "for all objects with a certain property such that some-
thing happens", then the method of specialization can be used. Without loss