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# ef11k - MAD 2104 Final Exam Dec 8 2011 Prof S Hudson 1[10...

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MAD 2104 Dec 8, 2011 Final Exam Prof. S. Hudson 1) [10 pts] How many bit strings of length 7 contain at least ﬁve 1’s ? 2) [10 pts] Prove ONE, in the usual paragraph style, including the phrase x more than once. Avoid shortcuts, such as quoting stronger theorems about sets (see me, if in doubt about this). A B A B If A B , then B A . 3) [5 points each] Let A = { 0 , 1 , 2 } . a) Give an example of a relation R on A , that is reﬂexive but not symmetric. You may describe R in words, or by a clearly-labeled matrix, or (di)graph. b) Give an example of a diﬀerent relation R on A , that is transitive but not an equivalence relation. If either part of this problem is impossible, explain why. 4) [10 points] Find a Boolean expression in sum-of-minterm form (eg DNF) for the function, F ( x,y,z ) = 1 if and only if x + y = 0. 5) [15 points] Answer True or False: If f : A A is onto, it must also be one-to-one. The wheel graph W n for n 3 is never bipartite. The complete graph K n for n 4 always has an even number of edges. The number of diﬀerent Boolean functions of degree n is always at least 4 n (for n 1). The terms of the Fibonacci sequence alternate between odd and even forever.

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ef11k - MAD 2104 Final Exam Dec 8 2011 Prof S Hudson 1[10...

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