lecnotes1 - Lecture Notes Concepts of Mathematics(21-127...

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Unformatted text preview: Lecture Notes, Concepts of Mathematics (21-127) Lecture 1, Recitation A–D, Spring 2008 1 Logic and Proofs 1.1 The Language of Mathematics In mathematics, a definition is used to give precise meaning to a word or phrase. These definitions describe mathematical concepts or properties that are of interest. The objec- tive of mathematics is to use definitions to produce theoretical results such as theorems, propositions, and lemmas. Here is a description of some types of results and related objects. • A theorem is usually a profound, sometimes complicated result that significantly adds to the mathematical theory of its subject. • A proposition is usually a somewhat lesser result, but still may be useful or interesting on its own. • A lemma is usually a basic result that might be used to help prove a proposition or theorem. • A corollary is something that follows almost immediately from another result. • An example is a specific case of a result, perhaps used to illustrate the significance of the result. • An algorithm is a procedure for solving a problem in a finite number of steps. A proof is a complete justification of the truth of a result. It generally begins with some hypotheses stated in the result and proceeds by logical deductions to the claimed statement. Along the way it may draw upon other hypotheses, previously defined concepts, previously proven results, or some basic axioms or postulates that are accepted as true. Q: How much detail should be included in a proof? A: Generally, the further you go in mathematics, the less detail and justification you will find given, as with more advanced topics more is expected of the reader. However, to develop good proof-writing skills, we will be complete and concise. 1 1.2 Logic Definition 1.1 A statement or proposition is a sentence or phrase that is either true (T) or false (F). Consider the following statements: 1. The Baltimore Orioles used to be the St. Louis Browns. 2. The only positive integers that divide 7 are 1 and 7 itself. 3. I will take my umbrella. 4. Minneapolis is the capital of Minnesota. 5. Where are my car keys? 6. Please watch your step. 7. x 2 = 36. 8. This sentence is false. The first four are propositions. Note that propositions (1) and (2) are true, (3) may or may not be true, and (4) is false. Items (5)–(8) are not propositions: (5) is a question, (6) is a command, the truth value of (7) depends on what x is, and (8) is a paradox as it cannot take on the value true or false. Propositions (1)-(4) are simple or atomic in the sense that they do not have any other propositions as components. Compound propositions can be formed by using connectives with simple propositions....
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This note was uploaded on 01/25/2010 for the course MATH 21127 taught by Professor Howard during the Spring '08 term at Carnegie Mellon.

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lecnotes1 - Lecture Notes Concepts of Mathematics(21-127...

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