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ECEN601-Chapter1 - Chapter 1 Logic and Set Theory To...

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Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea ,yourob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture .M a th em a t i c a llog i ci s the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is the Zermelo-Fraenkel set theory, together with the axiom of choice. Each of the axioms included in this the- ory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead tocontradictions .Avoid- ing such contradictions was one of the original motivations for the axiomatization of set theory. 1
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2 CHAPTER 1. LOGIC AND SET THEORY Arigorousanalysisofsettheorybelongstothefoundations of mathematics and mathematical logic. The study of these topics is, in itself,aform idab letask . For our purposes, it will suf±ce to approach basic logical concepts informally. That is, we adopt a naive point of view regarding set theory and assume that the meaning of a set as a collection of objects is intuitively clear. While informal logic is not itself rigorous, it provides the underpinning for rigorous proofs .T h er u l e sw ef o l l ow in dealing with sets are derived from established axioms. At some point of your academic career, you may wish to study set theory and logic in greater detail. Our main purpose here is to learn how to state mathematical results clearly and how to prove them. 1.1 Statements Aproofinmathematicsdemonstratesthetruthofcertain statement .I tistherefore natural to begin with a brief discussion of statements. A statement, or proposition , is the content of an assertion. It is either true or false, but cannot be both true and false at the same time. For example, the expression “There arenoclassesatTexas A&M University today” is a statement since it is either true orfalse.Theexpression “Do not cheat and do not tolerate those who do” is not a statement. Note that an expression being a statement does not depend on whether we personally can verify its validity. The expression “The base of the natural logarithm, denoted e ,isan irrational number” is a statement that most of us cannot prove.
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ECEN601-Chapter1 - Chapter 1 Logic and Set Theory To...

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