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# Beginning Logic MATH 10130

ND

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This 0 page Class Notes was uploaded by Briana Cassin on Sunday November 1, 2015. The Class Notes belongs to MATH 10130 at University of Notre Dame taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/232700/math-10130-university-of-notre-dame in Mathematics (M) at University of Notre Dame.

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Date Created: 11/01/15

Math 10130 Lecture 32 9 April 2008 Last time Last time Last time Last time Last time We considered quantified formulas in several different contexts Last Last time We considered quantified formulas in several different contexts 0 Example 3X Vy Mxy where Mxy is interpreted as X g y Last Last time We considered quantified formulas in several different contexts 0 Example 3X Vy Mxy where Mxy is interpreted as X g y true when the domain of discourse is the set of natural numbers N Last Last time We considered quantified formulas in several different contexts 0 Example 3X Vy Mxy where Mxy is interpreted as X g y true when the domain of discourse is the set of natural numbers N false when the domain of discourse is the set of integers Z Last Last time We considered quantified formulas in several different contexts 0 Example 3X Vy Mxy where Mxy is interpreted as X g y true when the domain of discourse is the set of natural numbers false when the domain of discourse is the set of integers Z 0 Principle Truth or falsity of a statement depends on the context Alternation of Quantlflers Alternation of Quantlflers Consider the following structure Alternation of Quantifiers Consider the following structure Domain of discourse people Alternation of Quantifiers Consider the following structure Domain of discourse people Mxy stands for X loves y Alternation of Quantifiers Consider the following structure Domain of discourse people Mxy stands for X loves y Q Do VX 3y Mxy and y VX Mxy express the same proposition Comparing the two formulas Comparing the two formulas VX 3y Mxy says For each person X there is a y such that X loves y Comparing the two formulas VX 3y Mxy says For each person X there is a y such that X loves y In other words Everybody loves somebody EN VX 3y Mxy says For each person X there is a y such that X loves y In other words Everybody loves somebody 3y VX Mxy says There is a person y such that for all persons X X oves y I mizgn karma Xi rr39 392 VX 3y Mxy says For each person X there is a y such that X loves y In other words Everybody loves somebody 3y VX Mxy says There is a person y such that for all persons X X loves y In other words Somebody is loved by everybody A tough example A tough example Domain of discourse the set of natural numbers N A tough example Domain of discourse the set of natural numbers N Mxy stands for X g y A tough example Domain of discourse the set of natural numbers N Mxy stands for X g y Let39s discuss this beast of a formula Domain of discourse the set of natural numbers N Mxy stands for X g y Let39s discuss this beast of a formula VX Vy Mxyamp x y Hz szamp x z ampszamp z I mizgfw Lawn A huge syntactic tree A huge syntactic tree VX Vy Mxyamp x y Hz szamp x z ampszamp z A huge syntactic tree VX Vy Mxyamp x y Hz szamp x z ampszamp z Vy Mxyamp x y Hz szamp x zampszamp z VX Vy Mxyamp x y Hz A Ixzamp x z ampszamp z Vy Mxyamp x y Hz szamp x zampszamp z Mxyamp x y Hz szamp x z ampszamp z y I 139 Lugrum VX Vy Mxyamp x y Hz A Ixzamp x z ampszamp z Vy Mxyamp x y Hz szamp x zampszamp z Mxyamp x y Hz szamp x z ampszamp z y Mxyamp x y Hz szamp x z ampszamp z y I miz Lugrum A hwgg gym ggm VX Vy Mxyamp x y Hz szamp x z ampszamp z Vy Mxyamp x y Hz szamp x zampszamp z Mxyamp x y Hz szamp x z ampszamp z y Mxyamp x y Hz szamp x z ampszamp z y szamp x z ampszamp z y I miz Lugrum A hwgg gym ggm VX Vy Mxyamp x y Hz szamp x z ampszamp z Vy Mxyamp x y Hz szamp x zampszamp z Mxyamp x y Hz szamp x z ampszamp z y Mxyamp x y Hz szamp x z ampszamp z y szamp x z ampszamp z y szamp x z szamp z y I miz Lugrum VX Vy Mxyamp x y Hz szamp x zampszamp z 6 Vy Mxyamp x y Hz szamp x zampszamp z Mxyamp x y Hz szamp x zampszamp z y Mxyamp xy 1 Hz szamp xzampszamp zY 5 szamp x z ampszamp z y 4 szamp x z 2 szamp z y 3 I miz Lugrum Breaking down the formula Breaking down the formula 1Mxyamp x y says X lt y Breaking down the formula 1Mxyamp x y says X lt y 2 szamp x 2 says X lt z Breaking down the formula 1Mxyamp x y says X lt y 2 szamp x 2 says X lt z 3 szamp z y says 2 lt y Breaking down the formula 1Mxyamp X y says X lt y 2 MXZamp X 2 says X lt z 3 szamp z y says 2 lt y 4 MXZamp X zampszamp z y says X lt z and z lt y or X lt z lt y

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