problem sheets to be answered as final exam will have similar questions. These would be very good to use for studying we have been told. ( need questions answered in a simplest way possible for me to understand also.) Problem Sheet for Week 1 Question 1: Suppose you have the set of natural numbers N together with the binary operation ? (subtraction). Is the set closed under the operation ? ?

Question 2: Suppose we have integers a,b,c,d (elements of Z) with the properties that a ? b and c ? d. Is it always true that ac ? bd ? Verify the inequality is true provided 0 ? a and 0 ? c

Question 3:Verify that (a2 ?b2) = (a?b)(a + b) (a3 ?b3) = (a?b)(a2 + ab + b2) where a,b ?R

Question 4: Find factorizations for the following values and expressions: 25 91 221 437 439 x2 + x?6 x2 ?5x + 6 x2 + 9

Question 5: Find the solutions to the following equations – assuming they exist. Try factorizing the expressions on the left of the equations x2 ?3x + 2 = 0 2×2 ?x?1 = 0

Question 6: Does the following cubic equation have a real-valued solution? Explain. (Do not try to calculate the solution exactly – just give reasons why you know there must be a solution) x3 ?11×2 + 3x?10

Question 7: Determine if the number 991 is a prime number. If the number is prime at what point can you stop looking for divisors (factors) of the number 991 ? What about in general ? 1

Question 8: Using Euclid’s Algorithm (EA) calculate the greatest common divisor of A = 21 and B = 171. Repeat for A = 221 and B = 247. Repeat for A = 133 and B = 69.

Question 9: A function f(N) is said to be ”order Nk ( written as O(Nk))provided lim?? f(N) Nk ? C for some positive value C and positive value k. Show that f(N) = N3 ?5N2 + 6N ?2 is O(N3) but not O(N2)

Question 10:The exponential function f(x) = ex dominates any ”power of x” function as x increases to in?nity. That is lim xk ex = 0 for every positive value k. Use the power series given below to verify this fact. ex = 1 + x + x2 2! + x3 3! + x4 4! + x5 5! + … its in pdf format…

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