logic_in_brief_part_01_8-22-2005 - LOGIC BY DOUG JONES...

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LOGIC BY DOUG JONES AUGUST 19, 2005 I. INTRODUCTION We’re going to learn some logical methods that will help us in math. The methods that we’ll study deal with understanding logical equivalences and proving elementary theorems. But we have to work up to it. ***** ***** ***** ***** ***** We know that math is easier to “do” using symbols rather than lots of words. For example, consider ( ) 53 5 1 xx 5 + =+ vs. “Five times an unknown quantity which has been increased by three is fifteen more than five times that unknown quantity.” Which one seems easier to comprehend? Analogously, it is easier to see the structure of, and deal with, logical operations when they have been put into symbolic form. 1 Also, an extremely important consideration in problem solving is the understanding and application of mathematical theorems 2 . For example, “If 2 x = , then 2 4 x = .” is a true statement (in effect a small theorem) which is frequently used in problem solving; however, its converse 3 , which is “If 2 4 x = , then 2 x = ,” 1 Both cases assume, of course, that we have a correct understanding of the meaning and use of the symbols involved. 2 Def. – Theorem: “A formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions….” (Merriam-Webster Online Dictionary copyright © 2005 by Merriam- Webster, Incorporated. http://www.m-w.com/cgi-bin/dictionary ). logic_in_brief.doc ……. . aug. 2005 ……. .page 1 of 36 . 3 We’ll discuss the converse later in these notes.
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and which is quite often misused in problem solving, is not a true statement. Thus it is important that we begin our study of logic with an investigation of the symbols and their proper usage – that is, the syntax of logic . We’ll also need some hard-and-fast definitions. II. DEFINITIONS Def. 1 : Statement : A statement is a sentence 4 that is either true ( T ) or false ( F ), but not both simultaneously. Ex. 1 5 : (a) Doug Jones is 6 ft. 9 in. tall. (b) 23 6 ×= Def. 2 : Open Sentence : An open sentence is a sentence with a variable in it such that for some (perhaps none) of the substitution values of the variable the sentence becomes a true statement and for other substitutions the sentence becomes a false statement. Ex. 2 : is an open sentence. 2 32 5 xx +− = 0 Def. 3 : Domain of a Variable : The domain of a variable in an open sentence is the set of all possible substitutions which “make sense” when substituted into the open sentence. Def. 4 : Solution Set : The solution set of an open sentence is the set of all values of the domain of the variable such that the resulting substitution yields a true statement . Ex. 3 : (a) “He was 42 nd President of the USA.” 6 Analysis : (i) Variable: “He” 4 This could be an English sentence, a German sentence, a Chinese sentence, or a mathematical sentence.
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This note was uploaded on 10/04/2011 for the course CALC 91449 taught by Professor Jones during the Spring '11 term at Delaware Tech.

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logic_in_brief_part_01_8-22-2005 - LOGIC BY DOUG JONES...

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