Advanced Mathematics

1.1Sentential Logic

Purpose of Section

To introduce sentential (sen‐TEN‐shuhl) or propositional logic and the fundamental idea of a sentence (or proposition) and show how simple sentences can be combined using the logical connectives “and,” “or,” and “not” to form compound sentences. We then analyze the meaning of these connectives by means of truth tables and then introduce the concepts of logical equivalence, tautologies, and contradictions. We close by introducing conjunctive and disjunctive normal forms of logical expressions.

1.1.1 Introduction

So what is mathematics? The word itself is derived from the Greek word mathēmatikē, meaning “knowledge” or “learning.” To both the practitioners of mathematics as well as the general public, the definition of the Queen of the Sciences varies widely.

• One of the greatest mathematician of Greek antiquity, Aristotle (384–322 BCE) defined mathematics as follows:

Mathematics is the science of quantity.

• Later, the Italian physicist Galileo (1564–1642), who was more interested in how it was applied, wrote:

“Mathematics is the language of the Universe and its characters are triangles, circles, and other geometric figures, without which it is impossible to understand a single word of it.”

• A more generic definition is given by the encyclopedia Britannica, which defines mathematics as follows:

Mathematics is the science of numbers and shapes and the relations between them.

• Then there is the beauty in mathematics as observed ...