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Rosen_SSG_Proofs - A Guide to Proof-Writing PW-1 A Guide to...

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A Guide to Proof-Writing PW-1 A Guide to Proof-Writing by Ron Morash, University of Michigan–Dearborn At the end of Section 1.7, the text states, “We have not given a procedure that can be used for proving theorems in mathematics. It is a deep theorem of mathematical logic that there is no such procedure.” This is true, but does not mean that proof-writing is purely an art, so that only those with exceptional talent and insight can possibly write proofs. Most proofs that students are asked to write in elementary courses fall into one of several categories, each calling for a systematic approach that can be demonstrated, imitated, and eventually mastered. We present some of these categories and techniques for working within them, organized as follows. This material supplements that found in the text and is intended to help get you started creating your own proofs. Also, studying the material in this Guide will help you understand better the proofs you read. 1. Deducing conclusions having the form “For every x , if P ( x ), then Q ( x ).” 1.1. Direct proof 1.1.1. Propositions having no hypothesis 1.1.2. Propositions having one or more hypotheses 1.1.3. Disproving false propositions having conclusions of the form “ x [ P ( x ) Q ( x )]” 1.1.4. The tactic of division into cases 1.1.5. Proving equality of sets 1.2. Indirect proof 1.2.1. Proof by contraposition 1.2.2. Proof by contradiction 1.2.3. Deriving conclusions of the form “ q or r 2. Remarks on additional methods of proof 2.1. Deducing conclusions having the form “For every x , there exists y such that P ( x, y ).” 2.2. Proof by mathematical induction 1. Deducing conclusions having the form “For every x , if P ( x ), then Q ( x ).” Many defining properties in mathematics have the form x [ P ( x ) Q ( x )], representing the idea “All P ’s are Q ’s.” (Cf. Examples 23, 26, and 27 in Section 1.3 of the text.) Some definitions involving this form are: (i) A set A is a subset of a set B : In symbols, A B if and only if x [( x A ) ( x B )]. This is read in words, “ A is a subset of B if and only if, for every x , if x A , then x B .” Less formally, A is a subset of B if and only if every element of A is also an element of B . (Cf. Definition 4 in Section 2.1 of the text.) (ii) A function f is one-to-one : f is one-to-one if and only if, for every x 1 and x 2 in the domain of f , if f ( x 1 ) = f ( x 2 ), then x 1 = x 2 . (Cf. Definition 5 in Section 2.3 of the text.) (iii) A relation R on a set A is symmetric : R is symmetric if and only if, for every x, y A , if ( x, y ) R , then ( y, x ) R . (Cf. Definition 4 in Section 8.1 of the text.) Many mathematical propositions that students are asked to prove have as their conclusion a statement involving a definition of the form just described. Some examples are: (a) Prove that for all sets A and B , A A B .
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