Problem_Set - PROBLEM SET Students are required to prepare...

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1 PROBLEM SET Students are required to prepare for answering questions about the problems before going to the tutorials. During tutorials, some students will be picked to show the answers of problems related to the lecture in that week. The discussion marks will be given to them based on their performance. WEEK 1-2 1. Write out the following formulae in English (how to read in English) Example: b a + : a plus b. a b a b a * b a / b a 2 a 3 a ) ( x f a b log a b a B a B a B a B A B A φ ' a " a
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2 * a b a = b a < b a > b a | | a 2. True or False: ) ( R Q P ¬ is equivalent to R Q P ) ( (T) 3. Write the negation of the statement “For some 2 > x , 5 2 < x ”: 4. True of False: The negation of the statement 9 , 3 2 x x is 9 , 3 2 > > x x . 5. If the value of Q P ¬ is false, what is the value of Q P ? 6. If the propositional function E ( p , t ) is “Person p does task t correctly,” write a proposition in symbols using quantifiers that expresses the idea “Nobody’s perfect.” WEEK 3-5 7. Prove that if 2 a is even then a is even. What style of proof did you use? 8. Prove that every natural number 10 > n can be written as a product of primes. 9. Prove that 1 1 .... 1 +++ ( n square roots, 2 n ) is an irrational number. 10. Suppose ) ( n P is a predicate on the natural numbers, and suppose N k ) 2 ( ) ( + k P k P . For each of the following assertions below, state whether (A) it must always hold, or (N) it can never hold, or (C) it can hold but need not always. Give a very brief (one or two sentence) justification for your answers. The domain of all quantifiers is the natural numbers. (a) 0 n ) ( n P (b) ) 2 ( ) 0 ( + n nP P (c) ) 1 ( ) 0 ( + n nP P (d) ) ( n P n ¬ (e) 10 ( n )) ( n P ( 10 > n ) ( n P ¬ ) (f) 10 ( n )) ( n P ¬ ( 10 > n ) ( n P )
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3 (g) ) 1 ( ) 0 ( P P ) ( n nP (h) n m n P > [ ) () ( m P ] (i) ) 1 ( ) ( + ¬ ¬ m P m P )) ( ( n mP n 11. Prove by induction that = = n i n i 2 1 1 1 12. Prove that 2 3 3 3 ) 2 1 ( 2 1 n n + + + = + + + L L for all integers n > 0. WEEK 6-8 13. Compute the following modular arithmetic: (2 * 3 * 4 – 1) mod 5 (0 – 3) mod 8 (2 + 79) 900000 mod 80 14. Solve the following equations for x 5 x + 23 6 (mod 499) 9 x + 80 2 (mod 81) 15. Compute gcd (11, 7) gcd (260, 320) 16. Compute the multiplicative inverse of 4 modulo 7 17. Follow the steps in the lecture to produce public key and private key. Let p = 2, q =11, e = 7. What are public key and private key ? If we use these keys to encrypt and
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Problem_Set - PROBLEM SET Students are required to prepare...

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