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# ef10k - MAD 2104 Final Exam and Key Prof S Hudson 1 How...

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MAD 2104 June 17, 2010 Final Exam and Key Prof. S. Hudson 1) How many ways can we arrange 5 men and 5 women in a row, if the men and women must alternate ? 2) Given the formula for the matrix A below, prove the one for A n using induction. The f n are the Fibonacci numbers [0, 1, 1, 2, 3, 5, etc] and you can use any well-known formula about them in your proof. A = 1 1 1 0 A n = f n +1 f n f n f n - 1 3) [7 points each] Recall that N is the set of non-negative integers. Give an example, using a formula if possible, of a function f : N N that is: a) one-to-one but not onto. b) onto, but not one-to-one (a different function from part a), of course). 4) [8 points each] Let A i = { 1 , 2 , 3 , 4 , . . . i } for i = 1 , 2 , 3 , . . . . Find these: a) n i =2 A i b) n i =3 A i 5) [20 points] Answer True or False: x [ P ( x ) Q ( x )] is logically equivalent to xP ( x ) ∧ ∀ xQ ( x ) x [ P ( x ) Q ( x )] is logically equivalent to xP ( x ) ∧ ∃ xQ ( x ) If R is an equivalence relation, then R 2 = R .

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