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Unformatted text preview: Fall 2007 ARE211 Problem Set #04 Answer key First Graphical Problem Set (1) Some quick multiple choice questions on necessary and sufficient conditions. (a) The most important question to ask in determining whether x is sufficent for y is: (i) Can you have y without having x ? (ii) Does x have any relevance for y ? (iii) Does having x ensure that you will have y ? (iv) If you know that you have y , do you know that you will have x ? Justify your answer using the fact that if x is sufficient for y then the set of objects satisfying x is contained in the set of objects satisfying y . Ans: The correct answer is (1(a)iii) . We know that if x is sufficient for y then the set of objects satisfying x is contained in the set of objects satisfying y . Therefore, having y implies having x . (b) The most important question to ask in determining whether p is necessary for q is: (i) Can you have p without having q? (ii) Does having p ensure that you will have q? (iii) Can p and q both be true at the same time? (iv) Do you have to have p in order to have q? Justify your answer using the fact that if p is necessary for q then the set of objects satisfying p contains the set of objects satisfying q . Ans: The correct answer is (1(b)iv) . We know that if p is necessary for q then the set of objects satisfying p contains the set of objects satisfying q . Therefore, if q is satisfied then p is satisfied, i.e., you have to have p in order to have q . 1 2 Ans: (2) Consider the following three class of triangles: (A) Isoceles (B) Equilateral (C) Rightangled Now answer the following questions (a) Which class of triangle is sufficient for one of the other classes? Explain your answer. (b) Which class of triangle is necessary for one of the other classes? Explain your answer. (c) Is it possible to identify a restriction on one of the angles in the remaining class of triangles such that if this additional restriction is satisfied, then this class is sufficient for another class? If yes, identify the condition. If no, explain in terms of set theory. (d) Is it possible to identify a restriction on one of the angles in the remaining class of triangles such that if this additional restriction is satisfied, then this class is necessary for another class? If yes, identify the condition. If no, explain in terms of set theory. Ans: (a) (B) is sufficient for (A): if a triangle has three equal sides, then it has two equal sides, and hence belows to (A); (b) (A) is necessary for (B): a triangle cannot have three equal sides without having two equal sides. (c) Right triangles with one 45 angle must have two sides equal. Hence, the class of Right angled triangles with a 45 angles is sufficient for the class of isoceles triangles....
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.
 Fall '07
 Simon

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