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Math 213 โ Practice exercises
September 20
Practice test 1
(1) Write the power set of {โ
, 1 , { 1 }}. Is { 1 } an element of this power set? (2) Consider the function f : N โ N given by f (n) = n^2 + n. Is f injective? What is the range (image) of f? Justify your answers. (3) Let A = {(m, n) โ N ร N|m + n = 6}, let B = {(m, n) โ N ร N|m, n โค 5 and m-n is even } and C = {(m, m)|m โ N}. Find A โ C, B โ C and (A
B) โ C.
(4) Let a and n be natural numbers and a โค n. Use induction on n to prove the following equality: โ^ n
i=a
i =
(n + a)(n โ a + 1) 2
(5) Let {an}nโฅ 1 be an arithmetic sequence such that a 3 = 8 and a 7 = 20. Find the general term an of the sequence. Then compute โ^24
n=
an.
(6) Compute the double sum:
i=
j=0(i^ โ^2 j). (7) Find the least integer n such that f (x) is O(xn) for f (x) = x^3 โ 99 x^2 + x^2 (log x)^4. (8) Let a > 1, b > 1 be two real numbers. Show that loga(x^4 + 7x^1 + 1) = ฮ(logb x). (9) Use the insertion sort algorithm to sort 2, 4 , 1 , 5 , 3.
1
Practice test 2
(1) Let A, B, C be three subsets of an universal set U such that 1100011010 is the bit string of A, 1011001000 is the bit string of B, 1010100101 is the bit string of C. Find the bit strings of A
B
C and (A
B)
C.
(2) Consider the function f : N โ Z given by
f (n) = (โ1)nb
n + 1 2
c.
Is f injective? Is it surjective? Justify your answers. (3) Let A = {x โ N| 3 โค x โค 21 , xโ 3 2 โ N}. Let B = {x โ N| 5 โค x โค 25 , |x โ 8 | โค 3 }, and C = {x โ N|x^2 โค 20 }. Find (A
C) โ B and A โ (B
C).
(4) Let a and n be natural numbers and a โค n. Let r 6 = 1. Use induction on n to prove the following equality: โ^ n
k=a
rk^ =
rn+1^ โ ra r โ 1
(5) Let {an}nโฅ 0 be a geometric sequence such that a 1 = โ3 and a 6 = 96. Find the general term an of the sequence. Then compute โ^12
n=
an.
(6) Compute the double sum:
i=
j=1(i^ โ^ 1)j. (7) Find the least integer n such that f (x) is O(xn) for f (x) = x(x โ 1)(x โ 2) log(x + 1). (8) Use the bubble sort algorithm to sort 2, 4 , 1 , 5 , 3.
