Let t be a positive integer. Prove that, if there exists a
Steiner triple system of...
Let t be a positive integer. Prove that, if there exists a
Steiner triple system of index 1 having v varieties, then there
exists a Steiner triple system having v^t varieties
Let A[1..n] be an array of distinct positive integers, and let t
be a positive integer.
(a) Assuming that A is sorted, show that in O(n) time it can be
decided if A contains two distinct elements x and y such that x + y
= t.
(b) Use part (a) to show that the following problem, re- ferred to
as the 3-Sum problem, can be solved in O(n2) time:
3-Sum
Given an array A[1..n] of distinct positive integers, and...
8.Let a and b be integers and d a positive
integer.
(a) Prove that if d divides a and d divides b, then d divides both
a + b and a − b.
(b) Is the converse of the above true? If so, prove it. If not,
give a specific example of a, b, d showing
that the converse is false.
9. Let a, b, c, m, n be integers. Prove that if a divides each of b
and c,...
Let n be a positive integer. Prove that if n is composite, then
n has a prime factor less than or equal to sqrt(n) . (Hint: first
show that n has a factor less than or equal to sqrt(n) )
Let F be a finite field. Prove that there exists an integer n≥1,
such that n.1_F = 0_F .
Show further that the smallest positive integer with this
property is a prime number.
6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (? − 1)?
+ ?? is ?(??+1).
7. Arrange the functions ?10, 10?, ? log ? , (log ?)3, ?5 + ?3 +
?2, and ?! in a list so that each function is big-O of the next
function.
8. Give a big-O estimate for the function ?(?)=(?3
+?2log?)(log?+1)+(5log?+10)(?3 +1). For the function g in your
estimate f(n) is O(g(n)), use a simple function g...
Let G be an abelian group and n a fixed positive integer. Prove
that the following sets are subgroups of G.
(a) P(G, n) = {gn | g ∈ G}.
(b) T(G, n) = {g ∈ G | gn = 1}.
(c) Compute P(G, 2) and T(G, 2) if G = C8 ×
C2.
(d) Prove that T(G, 2) is not a subgroup of G = Dn
for n ≥ 3 (i.e the statement above is false when G is...
7. Let m be a fixed positive integer.
(a) Prove that no two among the integers 0, 1, 2, . . . , m − 1
are congruent to each other modulo m.
(b) Prove that every integer is congruent modulo m to one of 0,
1, 2, . . . , m − 1.
(3) Let m be a positive integer. (a) Prove that Z/mZ is a
commutative ring. (b) Prove that if m is composite, then Z/mZ is
not a field.
(4) Let m be an odd positive integer. Prove that every integer
is congruent modulo m to exactly one element in the set of even
integers {0, 2, 4, 6, , . . . , 2m− 2}
Let {an} be a bounded sequence. In this question,
you will prove that there exists a convergent subsequence.
Define a crest of the sequence to be a
term am that is greater than all subsequent terms. That is,
am > an for all n > m
(a) Suppose {an} has infinitely many crests. Prove
that the crests form a convergent subsequence.
(b) Suppose {an} has only finitely many crests. Let
an1 be a term with no subsequent crests. Construct a...