N test2mad2104 page 3 of 4 4 15 pts a suppose f is a

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TEST2/MAD2104 Page 3 of 4 _________________________________________________________________ 4. (15 pts.) (a) Suppose f is a function defined recursively by f(0) = 1 and f(n+1) = -2f(n) for n = 0,1,2,3, .... Then f(1) = f(2) = f(3) = f(4) = (b) Give a recursive definition for the sequence {a n }, where n = 1,2,3,..., if a n = 6n. [Be very careful with the quantification used in the induction step.] Basis Step: Induction Step: (c) Give a recursive definition of the set S of positive integers that have a remainder of 2 upon division by 3. Be very careful with the quantification in the induction step. Basis Step: Induction Step: _________________________________________________________________ 6. (10 pts.) Use mathematical induction to prove that a set with n elements has n(n-1)/2 subsets containing exactly two elements whenever n is a positive integer greater than or equal to 2.
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TEST2/MAD2104 Page 4 of 4 _________________________________________________________________ 7. (10 pts.) Using a complete sentences, correctly name the fallacy exemplified by each of the following invalid arguments. (a) "If x is any real number with x > 3, then x 2 > 9. Suppose that x 3. Then x 2 9." (b) "If x is any real number with x > 3, then x 2 > 9. Suppose that x 2 > 9. Then x > 3." _________________________________________________________________ 8. (15 pts.) This is a valid argument: "All dogs are carnivorous. Some animals are dogs. Therefore some animals are carnivorous." The validity of the argument can be seen easily by symbolizing the argument using propositional functions and quantifiers as follows: Define propositional functions as follows: D(x) : "x is a dog." C(x) : "x is carnivorous." A(x) : "x is an animal." Then the argument translates into this: ( x)(D(x) C(x)) ( x)(A(x) D(x)) ( x)(A(x) C(x)) In the proof of validity which follows provide the missing justification(s) for each step. You should explicitly cite rules of inference and quantification, hypotheses, and preceding steps by number. [Note: The lower case latin letters ’a’,’b’,’c’ denote individuals in the universe of discourse.] 1. ( x)(D(x) C(x)) Justification: Hypothesis 2. ( x)(A(x) D(x)) Justification: Hypothesis 3. A(c) D(c) Justification: 4. D(c) C(c) Justification: 5. D(c) A(c) Justification: 3, is commutative. 6. D(c) Justification: 7. C(c) Justification: 8. A(c) Justification: 9. A(c) C(c) Justification: 10. ( x)(A(x) C(x)) Justification:
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