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Final Sample

# Final Sample - that is ﬁve cards that have consecutive...

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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. Prove that the relation a b (mod m ) is an equivalence relation on the set of integers. 4. If f g is one-to-one, does it follow that g is one-to-one? 5. What does it mean that a function f is in O ( g ) for some function g ? 6. 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 . 7. How many ways are there to choose a dozen donuts from the 21 varieties at a donut shop? 8. State Pascal’s identity. 9. What is the probability that a ﬁve-card poker hand contains a straight,
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Unformatted text preview: that is, ﬁve cards that have consecutive kinds? (Note that an ace can be considered either the lowest card of an A-2-3-4-5 straight or the highest card of a 10-J-Q-K-A straight.) 10. State the generalized pigeonhole principle. 11. State the Fundamental Theorem of Arithmetic. 12. Give a deﬁnition of an inverse of a number a modulo m . 13. Determine the greatest common divisor of 2689 and 4001....
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