Principles of Mathematics in Operations Research

4 1 Introduction

Needs proof! There is an angle t such that cos t = t.

Proof. Proofs should not contain ambiguity. However, one needs creativity, in-

tuition, experience and luck. The basic guidelines of proof making is tutored

in the next section. Proofs end either with Q.E.D. ("Quod Erat Demonstran-

dum"), means "which was to be demonstrated" or a square such as the one

here. •

Theorem 1.2.4 Theorems are important propositions.

Lemma 1.2.5 Lemma is used for preliminary propositions that are to be used

in the proof of a theorem.

Corollary 1.2.6 Corollary is a proposition that follows almost immediately

as a result of knowing that the most recent theorem is true.

Axiom 1.2.7 Axioms are certain propositions that are accepted without for-

mal proof.

Example 1.2.8 The shortest distance between two points is a straight line.

Conjecture 1.2.9 Conjectures are propositions that are to date neither proven

nor disproved.

Remark 1.2.10 A remark is an important observation.

There are also quantifiers:

3 there is/are, exists/exist

V for all, for each, for every

€ in, element of, member of, choose

3 such that, that is

: member definition

An example to the use of these delimiters is

~iy G S = {x e Z+ : x is odd }, y^2 e S,

that is the square of every positive odd number is also odd.
Let us concentrate on A => B, i.e. if A is true, then B is true. This
statement is the main structure of every element of a proposition family which
is to be proven. Here, statement A is known as a hypothesis whereas B is
termed as a conclusion. The operation table for this logical statement is given
in Table 1.1. This statement is incorrect if A is true and B is false. Hence,
the main aim of making proofs is to detect this case or to show that this case
cannot happen.