CHAPTER 1
Logic
Definitions
D1. Statements (propositions), compound statements.
D2. Truth values for compound statements p ∧ q, p ∨ q, p → q, p ↔ q. Truth
tables.
D3. Converse and contrapositive.
D4. Tautologies and contradictions. Logical equivalence.
D5. Quantifiers ∀ and ∃.
Facts (with proofs)
F1.
F2.
F3.
F4.
F5.
An implication and its contrapositive are logically equivalent statements.
(p ↔ q) ⇔ (p → q) ∧ (q → p)
De Morgan’s Laws.
Distributive Laws for conjunction and disjunction.
The negation of a statement involving one quantifier: ¬(∀xP (x)) ⇔
∃x(¬P (x)) and ¬(∃xP (x)) ⇔ ∀x(¬P (x)). Negation of the statements
involving several quantifiers.
1
2
1. LOGIC
Problems
Easier
#1.1 Let p and q and r be the propositions:
p = “m/3 is an integer”
q = “n/2 is an integer”
r = “(nm)/6 is an integer”
#1.2
#1.3
#1.4
#1.5
#1.6
#1.7
#1.8
Write the following statements using p, q, r and the symbols ∨, ∧, ¬, →
and ↔.
(a) “If m/3 is an integer and n/2 is an integer, then (nm)/6 is an integer”.
(b) The converse of your statement from part (a) above.
(c) The contrapositive of your statement from part (a) above.
(d) Is the statement in part (b) above true or false? Give your reasons.
Give truth tables for the following statements. Determine which are tautologies, which are contradictions and which are neither.
(a) ¬[(p ∧ ¬p) → q].
(b) ¬(q → p) → ¬p.
(c) (p ∧ q) ∧ ¬(p ∨ q).
(d) (p → q) → r.
(e) [(p → q) ∧ (q → r)] → ¬(p → r).
Suppose that the truth values of the statements p and r are true and the
truth values of the statements q and s are false. Find the truth values of
the following compound statements.
(a) ¬(¬(¬p ∨ q) ∨ r) ∨ s.
(b) (p → q) → (r → s).
Prove De Morgan’s Laws:
(a) ¬(p ∨ q) ⇔ ¬p ∧ ¬q.
(b) ¬(p ∧ q) ⇔ ¬p ∨ ¬q.
Prove that ¬(p ↔ q) is logically equivalent to ¬(p → q) ∨ ¬(q → p).
Let p and q be compound statements. If p is a contradiction, what can be
said about the truth value of statement p → q? If p is a tautology, what
can be said about the truth value of statement p → q? If q is a tautology,
what can be said about the truth value of the statement p → q?
Find the truth values of the following statements. Then construct their
negations.
(a) There exists an integer x such that (x2 − x − 1)(x + 1) = 0.
(b) For all real numbers x and y, x2 + y 3 ≥ 0.
Find the truth values of the following statements. Assume that all letters
represent real numbers.
(a) For all real x, (x2 − x − 1)(x + 1) = 0 implies x2 − x − 1 = 0.
(b) For all real numbers x, (x2 +1)(x3 −x−10) = 0 implies x3 −x−10 = 0.
2
2
(c) For all real numbers x, x
√ (x + 1) > 1 implies x(x + 1) > 1.
(d) For all real numbers x, 5x + 1 < 9 implies x < 16.
1. LOGIC
3
Medium
#1.9 Is the sentence “This statement is false” a proposition?
#1.10 Determine how many rows a truth table of a compound statement would
have if that involved 2,3,4 distinct simple statements, respectively. As
examples, consider compound statements ¬p ∧ ¬q, p ∨ q ∨ ¬r, (p ∨ q) →
(¬(r∧t))? You do not have to construct the complete tables to answer this
question. Do you see any pattern in your answers? What is the answer if
a compound statement is constructed out of n simple statements?
#1.11 Let P (x) and Q(x) be two predicates over reals (i.e., x is a real number).
Show that
(∃x such that P (x)) ∧ (∃x such that Q(x))
and
#1.12
#1.13
#1.14
#1.15
∃x such that (P (x)) ∧ Q(x))
are not necessarily logically equivalent.
Hint: It is sufficient to find examples of such P (x) and Q(x).
Let P (x, y) be the predicate x = 2y + 1, where x and y are integers. Find
the truth values of the following statement and explain your answer.
(a) ∃x∃y : P (x, y).
(b) ∃x∀y : P (x, y).
(c) ∀x∃y : P (x, y).
(d) ∀x∀y : P (x, y).
Determine whether the following statements are true or false. Assume
that x, y are reals. Prove your answers.
2
2
≥ xy).
(a) ∀x∀y ( x +y
2
x
(b) ∃x∃y ( y + xy < 2).
(c) ∃x > 0 ∃y > 0 ( xy + xy < 2).
Find the truth values of the following statements.
(a) For all real numbers x, x2 − 7x ≥ 12.
(b) For all real numbers x and y, x2 − 3xy + y 2 ≥ 0.
(c) For all real numbers x and y, x2 + xy + y 2 ≥ 0.
Let a, b, c be real numbers. Is the following statement correct? Prove your
answer.
∀a∀b∀c (a + b + c = 0 → a3 + b3 + c3 = 3abc).
#1.16 Suppose a, b, c are integers. Are the following statements true? Prove
your answers.
(a) ∀a∃b∃c (b2 + c2 = a2 ).
(b) ∀a 6= 0 ∃b 6= 0 ∃c 6= 0 (b2 + c2 = a2 ).
#1.17 In the following statement all letters represent real numbers:
∀M ∃N ∀x (x > N → x2 − 5x > M ).
Construct the negation of this statement. Decide whether the statement
or its negation is correct, and prove the correct statement.
4
1. LOGIC
Harder
#1.18 Is the following statement true?
For every integer n, there exist integers x and y such that
x2 − y 2 = 2n + 1.
Prove your answer.
#1.19 Prove that for all real numbers x, y, z,
x2 + y 2 + z 2 ≥ xy + xz + yz.
Moreover, prove that the equality sign is achieved if and only if x = y = z.
#1.20 Prove that for all real numbers x, y, z,
1 1
1
1
1
1
1
1
+ + =
implies 5 + 5 + 5 =
.
x y z
x+y+z
x
y
z
(x + y + z)5
#1.21 In the following statement all letters represent real numbers:
∀M ∃N ∀x (x > N → x sin x > M ).
Construct the negation of this statement. Decide whether the statement
or its negation is correct, and prove the correct statement.
1. LOGIC
5
Some Answers and Hints
Note that most of the comments below are hints. More details are needed when
you explain/proof your claims.
Easier
1.1
1.3
1.4
1.5
1.6
1.7
1.8
(1d) False.
(3a) False ; (3b) False
Use truth tables.
Use truth tables.
True; can be either True of False; True
(7a) True; (7b) False.
(8a) False; (8b) True; (8c) True; (8d) True.
Medium
1.9 No. If it is true, then it is false. If it is false, so it is true. This is an
example of how a self-reference may create a problem.
1.10 4; 8; 16; 2n .
1.12 (12a) True; (12b) False; (12c) False; (12d) False.
1.13 (13a) True; (13b) True; (13c) False.
1.14 (14a) False. Find a counterexample. (14b) False. Find a counterexample.
(14c) True. Complete the square.
1.15 Rewrite the first condition as a = −b − c.
1.16 (16a) True. Remember that 0 is an integer. (16b) False. Find a counterexample.
1.17 The negation can be constructed this way:
∃M ∀N ∃x ¬[(x > N → x2 − 5x > M )] ⇔
∃M ∀N ∃x ¬[¬(x > N ) ∨ (x2 − 5x > M )] ⇔
∃M ∀N ∃x (x > N ∧ x2 − 5x ≤ M )]
The original statement is correct, and, in our opinion, it is easier to prove.
Harder
1.18 Yes. If you have difficulty in finding a proof, let n be, e.g., 0, ±1, ±2, ±3, ±4, ±5.
In each of these cases find x and y as close to each other as possible. Try
to observe a pattern. Then generalize for an arbitrary integer n.
1.19 Use Problem 13a. (Make sure you know how to prove the statement you
are using).
1
imposes very strong restrictions on
1.20 The condition x1 + y1 + z1 = x+y+z
x, y, z. Try to understand what are they.
1.21 Remember that sin x takes value 0 for all x = πk, k ∈ Z.
CHAPTER 2
Set Theory
Definitions
D1. What do the notations N, Z, Q, R, C stand for?
D2. The empty set ∅, the power set P (A) of a set A, the Cartesian product
A × B of two sets A and B, the Cartesian n-th power An of a set A.
D3. Subsets, equality of sets, cardinality of a finite sets, operations on sets
(union, intersection, difference, symmetric difference).
D4. Universal set, complement of a set.
Facts
F1. Important Laws:
(a) De Morgan’s Laws: (A ∪ B)c = Ac ∩ B c and (A ∩ B)c = Ac ∪ B c
(with proofs).
(b) Distributive Laws: A∩(B ∪C) = (A∩B)∪(A∩C) and A∪(B ∩C) =
(A ∪ B) ∩ (A ∪ C) (with proofs).
F2. If |A| = m, |B| = n, then |A × B| = mn.
F3. If |A| = n, then |P (A)| = 2n (with proof).
7
8
2. SET THEORY
Problems
Easy
#2.1 Let A = {1, 2, 3, {1}, {1, 3}}. Which of the following statements are true
or false. Explain your answers.
(a) 1 ∈ A; 1 ⊂ A; {1} ∈ A; {1} ⊂ A.
(b) 2 ∈ A; 2 ⊂ A; {2} ∈ A; {2} ⊂ A.
(c) {1, 2} ∈ A; {1, 2} ⊂ A.
(d) {1, 3} ∈ A; {1, 3} ⊂ A.
(e) {1, 2, 3} ∈ A; {1, 2, 3} ⊂ A.
(f) |A| = 5.
#2.2 Let X1 = {M AT H210}, X2 = {M, A, T, H}, X3 = {210}, X4 = {M AT H},
and X5 = {2, 1, 0}. Find Xi ∩ Xj for i, j ∈ {1, 2, 3, 4, 5}.
#2.3 Suppose that the set X is a subset of Y. Find
(a) X ∪ Y
(b) X ∩ Y
(c) X \ Y
#2.4 Find the power set of the following sets:
(a) A = {∅, {∅}}.
(b) B = {x, y, z}.
#2.5 Determine if the following are elements of the set A × B × C where A =
{0, 1, 2}, B = {1, 2}, and C = {0, 1}.
(a) {∅}.
(b) {1}.
(c) (0, 2, 1).
(d) (2, 2).
(e) {(1, 1, 1)}.
#2.6 Given |A| = m, |B| = n and |C| = k determine the lower and upper
bounds for |(A × B) − C|.
#2.7 List all elements of the set {x ∈ R : 2x2 − x − 7 = 0}.
#2.8 Is the statement true or false? Explain.
(a) ∃m ∈ (0, 1] ∀x ∈ (0, 1] (m ≤ x).
(b) ∀x ∈ (0, 1] ∃y ∈ (0, 1] (x < y).
(c) ∀x ∈ (0, 1] ∃y ∈ (0, 1] (y < x).
(d) ∃M ∈ (0, 1] ∀x ∈ (0, 1] (x ≤ M ).
#2.9 Is the statement true or false? Explain.
(a) ∀x ∈ Z ∀y ∈ Z ∃z ∈ Z (x < y → x < z < y)
(b) ∀x ∈ Q ∀y ∈ Q ∃z ∈ Q (x < y → x < z < y)
#2.10 Given intervals of real numbers A = [1, 10], B = (10, 20], C = (5, 20).
Then, using set notation, A \ C can be expressed as {x ∈ R : 1 ≤ x ≤ 5}.
Express the following using set notation.
(a) A \ B.
(b) B \ C.
(c) A ∩ B.
(d) C \ B.
2. SET THEORY
9
Medium
#2.11
#2.12
#2.13
#2.14
#2.15
#2.16
#2.17
#2.18
#2.19
#2.20
#2.21
#2.22
#2.23
Prove that the empty set, ∅, is a subset of every set.
Let A = {a ∈ R : ∀x ∈ R (x2 + 6x ≥ a)}. Describe A.
Let C = {c ∈ R : ∀x ∈ R (x2 + 6x ≤ c)}. Describe C.
Let B = {b ∈ R : ∃x ∈ R (2x2 − bx + 3 = 0)}. Describe B.
Determine which of the following statements are true in the case of three
arbitrary sets P, Q, and R.
(a) If P is an element of Q and if Q is a subset of R, then P is an element
of R.
(b) If P is an element of Q and if Q is a subset of R, then P is also a
subset of R.
(c) If P is a subset of Q and Q is an element of R, then P is an element
of R.
(d) If P is a subset of Q and Q is an element of R, then P is a subset of
R.
Does A \ B = C imply A = B ∪ C? Prove your answer.
Does A = B ∪ C imply A \ B = C? Prove your answer.
Let A 6= ∅. Does A × B = A × C imply B = C? Prove your answer. What
if A = ∅?
Prove the following assertions involving three arbitrary sets A, B, and C.
(a) (A \ B) \ C = A \ (B ∪ C).
(b) (A \ B) \ C = (A \ B) \ (B \ C).
Prove (A \ B) ∪ (B \ A) = (A ∪ B) \ (A ∩ B) for every two sets A and B.
Prove that P (A) ∩ P (B) is equal to P (A ∩ B) for every two two sets A
and B. What can we say if the intersection operation, ∩, is replaced by
the union operation, ∪, in the above?
If the symmetric difference of sets A and B is equal to the symmetric
difference of the sets A and C, is it necessary that B = C? Explain your
answer.
Let Ai = {1, 2, 3, . . . , i} for i = 1, 2, 3, . . .. Find
n
n
\
[
Ai .
Ai and
i=1
i=1
#2.24 Let O be a point of a plane α. For each positive real number r, let
C(O, r) = {P ∈ α : OP = r}.
What geometrical figure C(O, r) is? Describe
[
C(O, r).
r∈R,r>0
#2.25 Let A = {1, 2, 3, . . . , 30}. For each integer i ≥ 2, define
Xi = {ik : k ∈ N and k ≥ 2}.
Describe the set
A\
30
[
i=2
Xi .
10
2. SET THEORY
#2.26 Prove the following assertions involving three arbitrary sets A, B, and C.
(a) A × (B ∩ C) = (A × B) ∩ (A × C).
(b) (A ∪ B) × C = (A × C) ∪ (B × C).
#2.27 A barber in an army was given an order to shave those and only those
men who do not shave themselves. Should he shave himself?
(This famous paradox shows that not every property can be used to
define a set.)
2. SET THEORY
11
Some Answers and Hints to Set Theory Section
Note that most of the comments below are hints. More details are
needed when you explain/proof your claims.
2.1. (a) T ; F ; T; T. (b) T; F; F; T. (c) F; T. (d) T; T. (e) F; T.
(f) T
2.2. E.g., X3 ∩ X5 = ∅, X2 ∩ X4 = ∅, X2 ∩ X2 = X2 . There are 15
different cases to consider.
2.3. (b) X
2.5. (c) Yes.
2.6. The lower bound is 0, the upper is mn. Think what C can be.
2.8. (a) F (c) T
2.9. (b) T
2.10. (c) ∅ (d) (5, 10]
2.12. a ≤ −9, or, equivalently, (−∞, −9]. Hint: Complete the square
in x2 + 6x, or find the y-coordinate of the vertex of the parabola y =
x2 + 6x.
2.13. C = ∅. √
√
2.14. b ≤ −2 6 or b ≥ 2 6.
2.15. (a) T (c) F
2.16. No. Give a counterexample.
2.17. No. Give a counterexample.
2.19. Can use either the method presented in the text, or the one
presented in class.
2.25 Hint: there are 11 numbers in the set.
CHAPTER 3
Relations
Definitions
D1. Relation from a set A to a set B. Relation on a set A.
D2. Domain and range of a relation.
D3. Reflexive, symmetric, antisymmetric and transitive relations. Equivalence
relation. Equivalence classes of an equivalence relation. Partition of a set.
D4. Functions. One-to-one (injective), onto (surjective), bijections (same as
one-to-one correspondence). Composition of functions. Inverse relation.
Inverse function.
D5. Cardinality of a set. Countable sets. Continuum.
Facts
F1. Equivalence classes of an equivalence relation on a set A partition A.
F2. Let f : A → B be a function. The inverse relation f −1 from B to A is
function if and only if f is a bijection. (with proof)
F3. Union and Cartesian product of two countable sets are countable. Q is
countable set. (with proof)
F4. Sets (0, 1)) and R are not countable. (with proof)
F5. For every set A, set P(A) has greater cardinality. (with proof)
Important Examples
E1. Examples of equivalence relations and the corresponding partitions of the
sets (equality of sets, equality of numbers, congruence of plane figures,
parallelism of lines, two integers having the same remainder when divided
by m.
E2. Relation which have exactly two of the properties of being reflexive, symmetric, or transitive, but not the third.
E3. Examples of functions which are (i) neither one-to-one nor onto; (ii) oneto-one but not onto; (iii) onto but not one-to-one; (iv) bijections.
13
14
3. RELATIONS
Problems
Easy
#3.1 Let R be the relation on N × N (where N is the set of natural numbers) by
((a, b), (c, d)) ∈ R if a ≤ c and b ≤ d. Determine whether R is reflexive,
symmetric, and/or transitive. Give reasons.
#3.2 Let α be a relation on N × N defined by the rule: ((a, b), (c, d)) ∈ α if
ad = bc. Prove α is an equivalence relation.
#3.3 Which of these collections of subsets are partitions of the set of integers?
Answer yes or no to each, and provide a reason.
(a) the set of negative integers and the set of positive integers.
(b) the set of integers divisible by 3, the set of the integers leaving a
remainder 1 when divided by 3, and the set of integers leaving a
remainder of 2 when divided by 3.
(c) the set of integers not divisible by 3, the set of even integers, and the
set of integers that leave a remainder of 3 when divided by 6.
#3.4 Find the range for the following functions defined over the integers (i.e.,
each function is from N to N). Assume that numbers are represented in
(the usual) decimal system. Explain your answers.
(a) the function that assigns each integer its last digit.
(b) the function that assigns each integer the number of digits in it.
#3.5 Give example of two sets A and B and a relation f from A to B such that
(a) f is onto, but not one-to-one;
(b) f is one-to-one, but not onto;
(c) domain of f = A, but f is not a function;
(d) range of f = B, but f is not a function.
#3.6 Prove that the function y = f (x) = 3x + 5 from Z to Z is not a bijection,
but a function given by the same formula from R to R is a bijection.
#3.7 What can be said about the cardinalities of two finite nonempty sets A
and B
(a) if there exists a function f from A to B which is one-to-one?
(b) if there exists a function f from A to B which is onto?
(c) if there exists a function f from A to B which is bijective?
(d) if there exists a function f from A to B?
Medium
#3.8 Let S = {x1 , x2 , x3 , x4 }. Let P (S) be the power set of the set S. Define the equivalence relation R on P (S) by (S1 , S2 ) ∈ R if and only if
|S1 | = |S2 |. Prove that R is an equivalence relation on P (S). How many
equivalence classes are there? Generalize to sets S = {x1 , . . . , xn }. How
many elements does each class have for n = 4?
#3.9 Let S be a set and suppose that x 6∈ S. Define the function f : P (S) →
P (S ∪ {x}) by f (A) = A ∪ {x} for all A ∈ P (S). Is this function f one-toone? Is this function f onto? Explain your answers. (P (S) denotes the
power set of S.)
#3.10 Is there a bijection between the set of all positive integers N and the set
of all squares of positive integers {1, 4, 9, 16, . . . }? Prove your answer.
3. RELATIONS
15
#3.11 Show by an example that it is possible to have a bijective function between
a set A and its proper subset B. (Hint: A must be an infinite set.)
#3.12 Let f : A → B and g : B → C be two one-to-one functions. Prove that
their composition g ◦ f is a one-to-one function from A to C.
#3.13 Let f : A → B and g : B → C be two functions, and let their composition
g ◦ f : A → C be one-to-one. Does it imply that both f and g are
one-to-one? Prove your answer.
#3.14 Let f : A → B and g : B → C be two one-to-one functions, and let their
composition g ◦ f : A → C be onto. Does it imply that both f and g are
onto? Prove your answer.
Harder
#3.15 Is there a bijection between the set of points of the whole real line and an
open segment (0, 1)?
#3.16 Find a bijection between (0, 1) and (0, 1].
#3.17 A circular disc in a plane is a union of a circle and its interior. Prove that
every infinite set of circular discs in plane such that no two discs share a
common point is countable.
16
3. RELATIONS
Some Answers and Hints to Relations Section
Note that most of the comments below are hints. More details are needed when
you explain/proof your claims.
3.1. Reflexive and transitive. Not symmetric.
3.2. It may remind the equality of two fractions a/b = c/d, but do not use it in
your arguments.
3.3. (a) No; (b) Yes (c) No.
3.4. (a) R(f ) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
3.5. One can use small sets A and B for all these examples.
3.7. (a) |A| ≤ |B|. (d) They can be arbitrary positive integers.
3.8. For n = 4 there are five equivalence classes. In general, i.e., for an arbitrary
positive integer n, there are n + 1 equivalence classes.
For n = 4 there are two equivalence classes with 1 element in each class, two equivalence classes with 4 elements in each class, and one equivalence class consisting of
6 elements.
3.15. Yes. There are several ways to present such bijection explicitly.
3.16. A part of your argument may use the idea similar to the one in ‘The infinite
hotel’ discussion held in class.
3.17. First try to explain that in any real interval (a, b), a < b, one can always find
a rational number.
CHAPTER 4
Mathematical Induction
Definitions
D1. Define “induction” or “inductive reasoning” in a broad sense.
D2. Define “deduction” or “deductive reasoning” in a broad sense.
D3. What is “the base of induction”, what is “the induction hypothesis”?
Facts
F1. State the Theorem of Mathematical Induction (both versions).
Important Examples
E1. Give examples of mathematical statements of the form “For all n ≥ n0 ,
P (n), which are correct for many initial values of n but false in general.
E2. Give examples of mathematical statements of the form “For all n ≥ n0 ,
P (n), where the implication P (k) → P (k + 1) is correct for all k ≥ n0 ,
but which are generally false due to the fact that P (n0 ) is false.
17
18
4. MATHEMATICAL INDUCTION
Problems
Prove the following statements by using Mathematical Induction.
Easier
#4.1
#4.2
Pn
#4.3
#4.4
#4.5
#4.6
#4.7
#4.8
1 · 2 · 3 + 2 · 3 · 4 + . . . + n(n + 1)(n + 2) = n(n+1)(n+2)(n+3)
.
4
xn+1 −1
2
n
1 + x + x + · · · + x = x−1 , x 6= 1.
Pn
(i + 1)2i = n2n+1 .
Pni=1
n(n+1)(4n−1)
·
i=1 (2i − 1)(2i) =
3
12 − 22 + 32 − 42 + . . . + (−1)n−1 n2 = (−1)n−1 n(n+1)
·
2
1 · 1! + 2 · 2! + 3 · 3! + . . . + n · n! = (n + 1)! − 1.
(By definition, 1! = 1, and n! = 1 · 2 · 3 · . . . · n (read “n factorial”).
Investigate for which positive integers n, n! > n2n , and prove your result.
Investigate for which positive integers n, 3n > 10n2 , and then prove your
result
Prove that for n ≥ 2, (1 + 1/3)n > 1 + n/3
Let x1 , x2 , . . . , xn be n real numbers, n ≥ 2. Prove that
#4.9
#4.10
#4.11
#4.12
j=1
+ 18
1
3
j 2 = 12 + 22 + 32 + . . . (n − 1)2 + n2 = n(n+1)(2n+1)
.
6
1
1
1
+ 15
+ 24
+ . . . + n2 +2n
= 34 − 2n22n+3
.
+6n+4
|x1 + x2 + · · · + xn | ≤ |x1 | + |x2 | + · · · + |xn |.
#4.13 Prove that for all n ∈ N, 5 | (n5 − n).
Medium
#4.14
1
1+x
+
2
1+x2
+
4
1+x4
+
8
1+x8
1
2n
2n+1
= x−1
+ 1−x
2n+1
1+x2n
sin n+1
nx
2 x
sin nx = sin x sin 2 , x 6=
2
+ ... +
|x| 6= 1.
#4.15 sin x + sin 2x + sin 3x + . . . +
2πk, where k
is an integer.
cos(n+1)x−1
#4.16 cos x + cos 2x + cos 3x + . . . + cos nx = (n+1) cos nx−n
, x 6= 2πk,
4 sin2 x
2
k is an integer.
#4.17 Let x1 , x2 , . . . , xn be n real numbers, n ≥ 2. Prove that
| sin(x1 + x2 + · · · + xn )| ≤ | sin x1 | + | sin x2 | + · · · + | sin xn |.
#4.18 (Bernulli’s Inequality) Prove that for any fixed real number x, −1 < x 6= 0,
and every integer n ≥ 2,
(1 + x)n > 1 + nx.
This inequality is useful to provide a simple rough bound on the values
of exponential functions. For example, what does it imply about the values of
1.0002100 or .99951000 ?
4. MATHEMATICAL INDUCTION
19
#4.19 Prove that for n ≥ 2,
1
1
1
+
+ ... +
> 1/2.
n+1 n+2
2n
This inequality leads to an easy proof of the famous fact that the sum
1/1 + 1/2 + 1/3 + . . . + 1/n can exceed any fixed number provided n being
sufficiently large.
#4.20 Prove that for every positive integer n,
1
1
1
1
1
+ 2 + 2 + ··· + 2 ≤ 2 − .
2
1
2
3
n
n
#4.21 Prove that for every positive integer n,
√
1
1
1
1
√ + √ + √ + · · · + √ > 2 n + 1 − 2.
n
1
2
3
#4.22 Prove that for n ≥ 2,
1 3 5
2n − 1
1
1
√ < · · · ... ·
<√
·
2 4 6
2n
2 n
3n + 1
#4.23 Prove that for every fixed a and b, a, b ≥ 0, and every integer n ≥ 2,
n
a+b
an + bn
≤
,
2
2
where the equality is attained if and only if a = b.
#4.24 Let a0 = 0, a1 = 3 and an = 23 (an−1 + 3an−2 + 3n ) for all n ≥ 2. Prove
that for all n ≥ 0, an = n3n .
√
#4.25 Consider a sequence {an }n≥1 , where a1 = 2 and an+1 = 2an + 5 for
n ≥ 1.
(i) Prove that an+1 > an for all n ≥ 1.
(ii) Prove that an < 4 for all n ≥ 1.
√
#4.26 Consider a sequence {bn }n≥1 , where b1 = 3 and bn+1 = bn + 2 for n ≥ 1.
(i) Prove that bn+1 < bn for all n ≥ 1.
(ii) Prove that bn > 2 for all n ≥ 1.
#4.27 Prove by induction that for all n ∈ N, 133 | (11n+2 + 122n+1 ).
#4.28 Prove that the sum of the cubes of three consecutive positive integers is
divisible by 9.
#4.29 Prove that for every integer n ≥ 1, there exist positive integers a and
b such that finding the gcd(a, b) by using the Euclidean algorithm takes
exactly n steps.
#4.30 3n + 1 coins are on a table, n ≥ 1. Two people play the following game:
in turn, they take either one or two coins from the table. The player who
gets the last coin looses the game. Prove that the second player has a
winning strategy, i.e., the second player can play the game in such a way
that he/she will always be a winner. (Of course, the decisions the second
player makes along the game may depend on how the first player has been
playing.)
#4.31 Consider the sequence of Fibonacci numbers defined as: F1 = 1, F2 = 1,
and Fn = Fn−1 + Fn−2 for all integers n ≥ 3. Prove that
(i) (Fn+1 )2 − Fn Fn+2 = (−1)n for all n ≥ 1.
(ii) F2 + F4 + F6 + . . . F2n = F2n+1 − 1
20
4. MATHEMATICAL INDUCTION
#4.32 Prove that every integer amount of n ≥ 18 dollars can be paid by using 4
or 7 dollar bills only.
√
√
#4.33 Show that for every positive integer n, (2+ 3)n = A+B
where A √
and
√ 3,
B are integers. Having done this, show that and (2 − 3)n = A − B 3,
where A, B are the same as in the first equality.
#4.34 Prove that the greatest number of regions that n ≥ 0 lines can divide the
plane is
n(n − 1)
.
1+n+
2
#4.35 Prove that the greatest number of regions that n ≥ 1 circles can divide
the plane is
n2 − n + 2.
#4.36 Consider a “map” formed on a plane by a finite number of lines, where the
“countries” are the regions formed by the lines. Prove that the countries
can be colored with just two colors such that any two countries that share
a common border (a segment) have different colors.
#4.37 Given an equal number of 0’s and 1’s distributed around the circle, show
that it is possible to start at some number and proceed around the circle
to the original starting position in such a way that, at any point during
the cycle, one has seen at least as many 0’s as 1’s.
#4.38 Let pn denote the n–th positive prime. Thus p1 = 2, p2 = 3, p3 = 5, p4 =
7, . . .. Prove that for n ≥ 5, pn > 2n.
Harder
#4.40 Consider a tournament which starts with n ≥ 1 teams. In the first round,
all teams are divided into pairs if n is even, and the winner in each pair
passes to the next round (no ties). If n is odd, then one random (lucky)
team passes to the next round without playing. The second round proceeds similarly. At the end, only one team is left – the winner. Find a
simple formula for the total number of games played in the tournament.
#4.41 Consider the sequence of Fibonacci numbers defined as: F1 = 1, F2 = 1,
and Fn = Fn−1 + Fn−2 for all integers n ≥ 3. Prove that every positive
for every n ≥ 1,
1
Fn = √ (αn − β n ) ,
5
√
√
1+ 5
1− 5
where α = 2 , and β = 2 .
#4.42 Prove that every fraction p/q, p and q are positive integers, p < q, can be
written in the form
1
1
1
1
p
=
+
+
+ +... +
,
q
n1
n2
n3
nk
where n1 , n2 , n3 , . . . , nk are positive integers satisfying n1 < n2 < . . . <
nk .
This way of representing fractions played an important role in Egyptian
mathematics.
4. MATHEMATICAL INDUCTION
21
#4.43 Prove that the greatest number of regions that n ≥ 0 planes can divide the
space is
n(n − 1)
n(n − 1)(n − 2)
1+n+
+
.
2
6
#4.44 Given an equal arm balance capable of determining only the relative weights of
two quantities and 3n − 1 coins, n ≥ 1, all of equal weight except possibly one
which is lighter. Show that it is possible to determine whether there is a light
coin and identify it in at most n weighings.
#4.45 (Little Fermat’s Theorem) Prove that for a given positive prime number p and
all integers a ≥ 0, p | (ap − a).
#4.46 Prove the Binomial Formula:
!
n
X
n n−i i
(a + b)n =
a
b,
i
i=0
` ´
` ´
where the binomial coefficients ni are defined by: n0 = 1, and
!
n · (n − 1) · . . . · (n − i + 1)
n
=
i
1 · 2 · ... · k
for 1 ≤ i ≤ n.
CHAPTER 5
Number Theory
Definitions
D1. When do we say that an integer b divides an integer a? What is a divisor,
and what is a multiple of an integer?
D2. When do we say that an integer b divides an integer a with remainder r?
D3. Congruence of two integers modulo integer n
D4. The greatest common divisor. Linear combination. Relatively prime integers.
D5. Prime integers, composite integers.
Facts
F1. The Well-Ordering Principle
F2. For all integers x, y, a, b, c, c 6= 0, if c|a and c|b , then c|(xa + yb) (with
proof).
F3. Division with Remainder Theorem. Proof of the uniqueness part.
F4. Be able to prove the following properties of congruences.
For any modulus m ∈ N, and all integers a, b, c, d, x, n, n ≥ 2,
(a) a ≡ a (mod m) (reflexive property)
(b) a ≡ b (mod m) ⇐⇒ b ≡ a (mod m) (symmetric property)
(c) If a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m)
(transitive property)
(d) a ≡ b (mod m) ⇐⇒ m|(a − b) ⇐⇒ a = mt + b for some t
(e) If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d
(mod m) and a − c ≡ b − d (mod m)
(f) If a ≡ b (mod m), and c ≡ d (mod m), then ac ≡ bd (mod m).
In particular, ac ≡ bc (mod m).
(g) If a ≡ b (mod m), then an ≡ bn (mod m)
(h) If a ≡ r (mod m) and 0 ≤ r < m, then r is the remainder of the
division of a by m.
F5. Be able to prove the following.
Let N = an−1 . . . a1 a0 be an n–digit positive integer, where a0 is the
number of units, a1 be the number of tens, and so on. Then
(a) N ≡ a0 + a1 + · · · + an−1 (mod 3)
(b) N ≡ a0 + a1 + · · · + an−1 (mod 9)
(c) N ≡ a0 − a1 + · · · + (−1)n−1 an−1 (mod 11)
(d) N ≡ a1 a0 (mod 4), where a1 a0 is the number formed by two
last digits of N
23
24
5. NUMBER THEORY
F6.
F7.
F8.
F9.
F11.
F12.
(e) N ≡ a2 a1 a0 (mod 8), where a2 a1 a0 is the number formed by
three last digits of N
Prove that the Euclidean algorithm gives the gcd(a, b).
Prove that if d = gcd(a, b), then there exist integers u and v such that
d = ua + vb.
Prove that
(a) if c|ab and gcd(a, c) = 1, then c|b
(b) if gcd(a, c) = gcd(b, c) = 1, then gcd(ab, c) = 1
(c) if a|c, b|c and gcd(a, b) = 1, then ab|c
(d) gcd(a, b) = d ⇐⇒ gcd(a/d, b/d) = 1
(e) if gcd(c, n) = 1, then a ≡ b (mod n) ⇐⇒ ac ≡ bc (mod n),
i.e., both sides of a congruence can be multiplied or divided by
an integer relatively prime to n.
Statement and proof of the Prime Factorization Theorem.
Prove that for every positive integer n, there exist n consecutive composite
integers.
Prove the following criterion for verifying
whether the integer is prime:
√
If n ≥ 2 and no prime number p ≤ n divides n, then n is prime.
CHAPTER 6
Combinatorics
Definitions
D1. Power set P(A) of a set A.
D2. k-permutations of a set A. Permutations of a set A. k-subsets (same as
k-combinations) of a set
A.
D3. Binomial coefficient nk .
Facts
F1. If Ai is finite for all i = 1, 2, . . . , k, then
|A1 × A2 × . . . × Ak | = |A1 | · |A2 | · . . . · |Ak |.
F2. Let |A| = n. The number of all subsets of A is 2n (with proof).
F3. The number of all binary relations from A to B, |A| = m, |B| = n, is 2mn
(with proof).
F4. Inclusion-Exclusion Principle.
F5. Let |A| = m and |B| = n.
(1) The number of all functions from A to B is nm (with proof).
(2) The number of all one-to-one functions from A to B is n · (n − 1) · . . . ·
(n − m + 1) (with proof).
(3) The number of all k-permutations of A is m · (m − 1) · . . . · (m − k + 1) =
m!
(m−k)! , and the number of permutations of A is m! (with proof).
m!
(4) The number of all k-subsets of A (same as k-combinations of A) is (m−k)!k!
(with proof).
Pn
(5) |P(B)| = k=0 nk = 2n (with proof). Pn
(6) Binomial formula: (a + b)n = k=0 nk an−k bk (with proof).
25
26
6. COMBINATORICS
Problems
Easy
#6.1 How many groups of 12 people which contain 4 men, 5 women and 3
children can be chosen from all people in an apartment building, if 22
men, 20 women and 28 children live there?
#6.2 A four-digit number is a number which is represented in decimal system
by four digits with the first digit not equal to zero. How many four-digit
numbers can be made out of digits 0, 1, 2, 3, 8, 9 if
(a) the digits in a number can repeat;
(b) no digit can repeat;
(c) the number is odd and digits can repeat;
(d) the number is odd and digits can not repeat.
#6.3 Twenty points are taken on the circumference of a circle. Any two of them
are endpoints of a chord. How many chords are there?
#6.4 How many integers in the set {1, 2, . . . , 1000} are
(a) divisible by 5? divisible by 3?
(b) divisible by 15 ?
(c) divisible by 5 or by 3?
(d) divisible by neither 3 nor 5?
(e) divisible by only one of the numbers 3 or 5?
2 19
#6.5 What is the coefficientP
at x5 y 28
in then expansion of (2x − 3y ) ?
n
n i
#6.6 Prove that for n ∈ N, i=0 i 4 = 5 .
Medium
#6.7 Given 2n points on a plane. One wants to draw n segments which join
pairs of these points such that no two segments share a common vertex.
In how many ways this can be done?
#6.8
(a) Expand 0 = (1 − 1)n by using the Binomial Theorem.
(b) Prove that for every n ≥ 1,
n
n
n
n
n
n
+
+
+ ··· =
+
+
+ · · · = 2n−1 .
0
2
4
1
3
5
(c) Show that there exist as many subsets of an n-element set which
contain an odd number of elements as there are those which
contain an even number of elements.
#6.9 What is the term with the largest coefficient in the expansion of
(2x + 3y)100 ?
#6.10 Given sets A and B, |A| = s ≥ 1, |B| = t ≥ 1.
(a) How many relations from A to B are there?
(b) How many relations from A to B have their domain consisting
of exactly one element of A ?
(c) How many functions from A to B are there?
(d) How many 1-to-1 functions from A to B are there?
(e) How many functions from A to B are bijections?
6. COMBINATORICS
27
(f) How many functions from A to B have their range consisting of
exactly one element of B (constant functions)?
(g) For how many relations φ from A to B the following is true:
∀a ∈ A ∃b ∈ B (aφb) ?
(h) For how many relations φ from A to B the following is true:
∃a ∈ A ∃b ∈ B (aφb) ?
(i) For how many relations φ from A to B the following is true:
∀a ∈ A ∀b ∈ B (aφb) ?
#6.11 Each of 7 radio operators from city A wants to establish a connection with
every one of 5 radio operators from city B.
(a) How many distinct connections between these two groups can be
established?
(b) In how many connections between the groups every person from
A is connected to
(i) exactly one person from B;
(ii) at least one person from B?
(iii) at most one person from B?
#6.12 Let A and B be two different people in a group of twelve. Suppose these
twelve people form a line to the cashier’s office. In how many of these
lines
(a) A stands immediately in front of B?
(b) A and B stand next to each other?
(c) There are exactly three people between A and B?
(d) A stands somewhere before B?
#6.13 A four-digit number is a number which is represented in decimal system
by four digits with the first digit not equal to zero. How many four-digit
numbers can be made out of digits 0, 1, 2, 3, 8, 9 if the number if even and
digits do not repeat.
#6.14 Let n = pqr, where p, q, r are distinct positive prime numbers. How many
integers from {1, 2, . . . , n} are relatively prime with n?
#6.15 How many sequences of length seven made out of letters A, C, G, U are
there with the property that a sequence coincides with itself if the order
of the letters is reversed?
#6.16 How many distinct “words” can be made by rearranging letters in the
word MATHEMATICS? In the word MISSISSIPPI?
#6.17 How many diagonals does a polygon with n ≥ 3 vertices have? (A diagonal
is the segment joining two vertices of the polygon and which is not a side
of the polygon.)
#6.18 How many graphs with a given set of n vertices are there?
#6.19 How many graphs with a given set of n vertices and m edges are there?
#6.20 Let N = pe11 pe22 . . . pekk , where all pi are distinct primes, and all ei are
nonnegative integers. How many positive distinct divisors does N have?
#6.21 In how many ways 12 identical coins can be placed in 4 distinct wallets
such that no wallet is empty?
28
6. COMBINATORICS
#6.22 How many solutions (x1 , x2 , . . . , x6 ) (ordered 6-tuple) does the equation
x1 + x2 + · · · + x6 = 50
have if
(a)
(b)
(c)
(a)
all xi ’s are nonnegative integers?
all xi ’s are positive integers?
all xi ’s are positive integers and each is at least 4?
A man has 10 distinct candies and he puts them in two distinct
bags such that each bag contains 5 candies. In how many ways
can he do it?
(b) A person has 10 distinct candies and he puts them in two identical bags such that no bag is empty. In how many ways can he
do it?
(c) A person has 10 identical candies and he puts them in two identical bags such that no bag is empty. In how many ways can he
do it?
#6.23 A woman has 6 friends. Each evening, for 5 days, she invites 3 of them
so that the same group is never invited twice. How many ways are there
to do this? (Assume that the order in which groups are invited matters.)
#6.24 In how many ways 14 distinct books can be placed on 5 distinct shelves
if the order of books on a shelf matters?
HARDER
#6.25 In how many 0-1 sequences of length 15 no two 1’s are next to each other?
#6.26 A four-digit number is a number which is represented in decimal system
by four digits with the first digit not equal to zero. What is the sum of
all four-digit numbers which can be made out of digits 0, 1, 2, 3, 8, 9 such
that no digit repeats?
#6.27 Fifteen points are taken on the circumference of a circle. Through any
two of them a chord is drawn. If no three of the chords intersect at a
point inside the circle, how many points of intersections of these chords
are there (including points themselves)?
#6.28 Suppose that all streets in a city form a rectangular grid with 5 (k) horizontal streets and 8 (n) vertical streets. In how many ways one can walk
from the South-West corner to the North-East corner if the only directions
one can walk are North and East?
#6.29 An international committee consists of 9 (or n) members. Committee
materials are stored in a safe. How many lock should the safe have, and
how many keys should be made for these locks, and how these keys have
to be distributed among the committee members such that the safe can
be open if and only if at least 6 (or k) members of the committee are
present?
#6.30 What is the greatest number of regions that n planes can divide the space?