Introcution to Discrete Mathematics
Introcution to Discrete Mathematics CIS 275
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This 3 page Class Notes was uploaded by Lindsay Bergstrom Sr. on Wednesday October 21, 2015. The Class Notes belongs to CIS 275 at Syracuse University taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/225605/cis-275-syracuse-university in Computer & Information Science at Syracuse University.
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Date Created: 10/21/15
I received a question about the homework assignment that is due tomorrow For 36 the chapter doesnt really give you a 7starting equation7 to set equal to AB l was wondering should we assume that the expression in question 35 is correct and use that as a starting equation for 367 This question is particularly good because it combines several important issues First a mathematical de nition is just a de nition that is logically rigorous and unambiguous Often such a de nition is most easily given using previously de ned symbols so that the de nition ends up looking mathematical But that7s not what would make it or fail to make it a mathematical de nition Rosen gives his readers the de nition of 69 just prior to exercise 32 of section 22 edition 6 He says The symmetric difference of A and B denoted by A 69 B is the set consisting of those elements in either A or B but not in both A and B This de nition does not look like an equation to start from and manipulate But using mathematical notation we can restate the de nition as A EBB x l m is in either A or B but not in both A and B The notation l 7 7 7 is read the set of all such that 7 7 7 So x l x is in either A or B but not in both A and B reads the set of all x such that z is in either A or B but not in both A and B The mathematical notation is effectively just a change of notation from somewhat peculiar English to peculiar English augmented with mathematical symbols it is not a deductive step Look at Rosen7s De nition 1 where he de nes the union of two sets He says The union of the sets A and B denoted by AU B is the set that contains those elements that are either in A or in B or in both Now with the convention that or as well as either or77 allows for both in mathematical discourse Rosen7s use of the phrase or in both77 is only for emphasis Strictly speaking it is redundant So we could restate his de nition as The union of the sets A and B denoted by AU B is the set that contains those elements that are either in A or in B Thus by this de nition of the union of two sets we can restate the set of all x such that z is in either A or B but not in both A and B as the set of all x such that z is in A U B but not in both A and B In turn using Rosen7s de nition of the intersection of two sets we can re eXpress this description of a set as the set of all x such that z is in A U B but not in A B And then using Rosen7s notation z E A that says z is an element of A we can re eXpress z is in A U B as x E A U B and re eXpress but z is not in both A and B as x Z A B We therefore obtain A Bzlx AUBbutz ZA B Now translating the previous line into the notation of propositional logic but where atoms are allowed to be simple sentences and not just Boolean variables we get A Bxlz AUB x A B Finally by Rosen7s de nition of the difference of two sets we get AEBBAUB7A B That7s exercise 35 You should note in working through the above argument that every step except the last one was just a matter of translation from one notation ordinary7 but peculiar English to another more concise notation No deductions were involved The last step did involve a deduction by combining the equation in Rosen7s de nition of the difference of two sets with our expression sleAUB x A B It is extremely important to realize that there is nothing special about mathematical proof Format and notation are not essential A mathematical proof is just a logically rigorous argument Sometimes7 but not always7 the easiest way to give such an argument is by doing or including a calculation or manipulating a mathematical expression But7 you will have a breakthrough when you de emphaisze that kind of thing in your own thinking7 and instead emphasize logically rigorous argumentation Don7t think in terms of trying to catch on to this material by trying to imitate manipulations of mathematical expressions That wont get far In doing exercise 367 you may assume the result of exercise 35 Exercise 36 will then be very7 very much shorter Howard Blair
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