DISCRETE MATHEMATICS, TEST 2 PRACTICE
(1) Prove or disprove the following statement: If r ∈ R and r3 is irrational then r is
irrational.
(2) Let n be any integer not divisible by 5. Prove or disprove the following statement:
{n2 mod 5 = 1} ∨ {n2 mod 5 = 4}.
(3) Prove from first principles that the number
√
2+
√
3 is irrational.
(4) Prove that if n is an integer greater than 5 such that both n − 1 and n + 1 are
prime, then n is divisible by 6.
(5) Use mathematical induction to prove that for all integers n ≥ 3,
n
X
(n − 2)(n2 + 2n + 3)
(j − 1)j =
3
j=3
(6) Use mathematical induction to prove that for all integers n ≥ 0,
n Y
1
1
1
.
=
2i + 1 2i + 2
(2n + 2)!
i=0
(7) Prove that for every integer n ≥ 1,
n
2
X
1
i=2
i
≥
n
2
Use this statement to comment on the convergence of the harmonic series
∞
X
1
i=2
i
2
DISCRETE MATHEMATICS, TEST 2 PRACTICE
(8) Use strong mathematical induction to prove that for integers n ≥ 2, either n is
prime or n is a product of prime numbers.
(9) Consider the Fibonacci numbers defined by the recurrence relation
Fn = Fn−1 + Fn−2 , n ≥ 3
and F1 = 1, F2 = 1. Prove by induction that for all n ≥ 1, F3n is always even.
(10) In a Double Tower of Hanoi with Adjacency Requirement there are three poles in a
row and 2n disks, two of each of n different sizes, where n is any positive integer.
Initially pole A (at one end of the row) contains all the disks, placed on top of each
other in pairs of decreasing size. Disks may be transfered one-by-one from one pole
to an adjacent pole and at no time may a larger disk be placed on top of a smaller
one. However a disk may be placed on top of one of the same size. Let C be the
pole at the other end of the row and let sn = {the minimum nimber of moves to
transfer a tower of 2n disks from pole A to pole C}.
a) Determine s1 and s2 .
b) Find a recurrence relation expressing sk in terms of sk−1 for all integers k ≥ 2.
Justify you answer carefully.
(11) A sequence c0 , c1 , c2 , . . . is defined as follows
c0 = 1, ∀k ≥ 1, ck = 7ck−1 + 2
Guess and then prove an explicit formule for the sequence {ci }.
(12) A certain computer algorithm excecutes twice as many plus one operations when it
runs with an input of size k as when it runs with input of size k − 1 (where k ∈ N).
When the algorithm is run on input of size 1, it executes 9 operations. How many
operations does the algorithm run when it is executed on an input of size n?
(13) Prove the following statement using an element argument: For all sets A, B and C
A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ C.
(14) Derive the following result algebraically using the Boolean algebra identities. Give
a reason for every step
(A ∪ C)\B = (A\B) ∪ (C\B).
DISCRETE MATHEMATICS, TEST 2 PRACTICE
3
(15) Prove that for all n ≥ 1, if A and B1 , B2 , . . . are any sets, then
∩ni=1 (A × Bi ) = A × (∩ni=1 Bi ) .
(16) For any sets A and B prove that
P(A) ∪ P(B) ⊆ P(A ∪ B).
Could the subset inclusion be proper? Give an example.
(17) Consider sets A, B and C. Simplify the following expression. Cite a Boolean
algebra identity at every step.
((A ∩ (B ∪ C)) ∩ (A\B)) ∩ (B ∪ C c ).
(18) Prove that the set Z2 is countably infinite.
(19) Let P(S) be the set of all substes of a set S, and let T be the set of all functions
from S to the set {0, 1}. Prove that P(S) and T have the same cardinality.
(20) Imagine that there is a hotel, called Hilbert Hotel, that contains a countably infinite
number of rooms. The nice featute of this hotel is that even when it is full, it can
always take another guest. For convenience suppose that the rooms are numbered
1, 2, 3, . . . and G is the set of guests. If the hotel is full then we have a bijection g :
N → G. Suppose that a new guest z arrives. Find a new bijection h : N → G ∪ {z}
to accomodate the new guest.