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# fexam-sample - G = V,E is n-regularfor some positive...

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Final Exam on MACM-101 Discrete Mathematics 1. What is an open variable? 2. Prove that sets A and B are disjoint if and only if A B = A Δ B . 3. If f g is one-to-one, does it follow that g is one-to-one? 4. What does it mean that a function f is in O ( g ) for some function g ? 5. Prove that for every positive integer n 1 · 2 1 + 2 · 2 2 + 3 · 2 3 + ... + n · 2 n = ( n 1)2 n +1 + 2 . 6. How many ways are there to choose a dozen donuts from the 21 varieties at a donut shop? 7. State Pascal’s identity. 8. Prove that if every cycle in a graph has even length then the graph is bipartite. 9. Show that if a bipartite graph (not necessarily complete!)
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Unformatted text preview: G = ( V,E ) is n-regularfor some positive integer n and ( V 1 ,V 2 ) is a bipartition of V , then | V 1 | = | V 2 | . That is, show that the two sets in a bipartition of the vertex set of an n-regular graph must contain the same number of vertices. 10. State Kuratowski’s Theorem. 11. State the Fundamental Theorem of Arithmetic. 12. Give a de±nition of an inverse of a number a modulo m . 13. Determine the greatest common divisor of 2689 and 4001....
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