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Unformatted text preview: 1 Definitions and Mathematical Terminology 1.1 Welcome Welcome to the third module in our course on proofs, Definitions and Mathematical Terminology. Before going on, please do the assigned readings. 1.2 Introduction If mathematics is thought of as a language, then definitions are the vocabulary and our previous mathematical knowledge indicates our experience and versatility with the language. 1.3 Definitions A definition in mathematics is an agreement, by all parties concerned, as to the exact meaning of a particular term. You have already seen the definition of “ A ⇒ B ” and that definition was communicated by a truth table. You do not have to accept the definition, but in refusing to accept a definition you will not be able to communicate with other mathematicians. Mathematics and the English language both share the use of definitions as extremely practical abbreviations. Instead of saying “a domesticated carnivorous mammal known scientifically as Canis familiaris ” we would say “dog.” Instead of writing down a truth table, we would say “ A implies B .” However, mathematics differs greatly from English in precision and emotional content. Mathe- matical definitions do not allow ambiguity or sentiment. Here are some examples of definitions. • An integer n divides an integer m , and we write n | m , if there exists an integer k so that m = kn . • An integer p > 1 is prime if the only positive integers that divide p are 1 and p (note that this definition makes use of the first definition). • A triangle is isosceles if two of its sides are equal in length. • An integer n is even if and only if the remainder when n is divided by 2 is 0. Definitions apply not only to the expected mathematical objects, algebraic objects and geometric objects, but it also applies to statements. • Two statements A and B are equivalent if and only if “ A implies B ” and “ B implies A .” • The statement “ A AND B ,” written A ∧ B (where “and” is an upward pointing angle), is true if and only if A is true and B is true. • The statement “ A OR B ,” written A ∨ B (where “or” is a downward pointing angle) is true in all cases except where A is false and B is false. Note that in all cases the definition stands as an abbreviation for a mathematical expression. The fact that it is a definition is sometimes indicated by “if and only if,” sometimes by “if,” and sometimes by “is” or “are.” The context is normally clear....
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- Spring '08