Test your understanding of MATH 135
To get more ready for MATH 145, I recommend reading the MATH 135 course notes and making
sure you can answer the following questions. They are mostly taken from Fall 2022 MATH 135.
1
Quantifiers
1.1 Express the following statements symbolically. Do not use any words.
(a) A perfect square is always non-negative.
(b) The range of the function y = x3 is all real numbers.
1.2 State the negation of the following statements without using words or the ¬ symbol. Determine if they
are True or False.
√
(a) ∀x ∈ Z, ∃y ∈ R, x + 2y < 3 2
(b) ∃a ∈ N, ∀b ∈ Q, ∃c ∈ Z, ¬(a = b − c)
1.3 For each sentence, determine whether or not it is true or false.
2
(a) ∃k ∈ Z, 8k (4k ) = 2
(b) ∀x ∈ R, x2 − x +
1
>0
4
(c) ∀x ∈ {0, 1, 2, 3}, ∀y ∈ {0, 1, 2, 3},
x+y
∈Q
x2 + y 2
(d) ∀x ∈ N, 4x + (ln x)2 ≥ 2x ln(x2 )
(e) ∀y ∈ Q, ∃x ∈ Q, x + 2xy = 4
1.4 Choose the appropriate domain, S, and the appropriate open sentence, P (x, y), to make each statement
true. In your solution, you must use each domain and each open sentence exactly once. You do not need
to provide proofs that these statements are true.
Domain, S
Open sentence, P (x, y)
{1, 2, 3}
x≤y
4680
∈Z
N
x2 + y 2
Q
x2 + y 3 = 135
R
x3 y 2 = 108
1
2
(a) ∀x ∈ S, ∀y ∈ S, P (x, y). Where S :
, and P (x, y) :
.
(b) ∀x ∈ S, ∃y ∈ S, P (x, y). Where S :
, and P (x, y) :
.
(c) ∃x ∈ S, ∀y ∈ S, P (x, y). Where S :
, and P (x, y) :
.
(d) ∃x ∈ S, ∃y ∈ S, P (x, y). Where S :
, and P (x, y) :
.
Logic
2.1 Consider the implication S(x, y): If xy < 0, then x > 0 and y < 0.
(a) State the hypothesis of S(x, y).
(b) State the conclusion of S(x, y).
(c) State the converse of S(x, y).
(d) State the contrapositive of S(x, y).
(e) State the negation of S(x, y) in a form that does not contain an implication.
(f) Determine the truth value of ∀x ∈ R, ∀y ∈ R, S(x, y).
2.2 Let A, B, and C be statement variables. Use a truth table or the laws of boolean algebra to prove that
(a) ((A =⇒ B) =⇒ C) ≡ (C ∨ A) ∧ (B =⇒ C).
(b) [A ∨ ¬(¬B ∨ C)] ≡ [(C =⇒ A) ∧ (A ∨ B)].
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Proofs
3.1 Two statements are given below. For each, an erroneous proof is provided. There is one main error
in each proof. State what this error is and explain why it is an error. (Note that the first of these
statements is true and the second is false, but this is irrelevant to the question and your answer.)
(a) Statement: For all real numbers x such that cos x ̸= 0 and sin x ̸= 0,
tan x + cot x = sec x csc x.
sin x
,
Proof: We are aiming to prove that tan x + cot x = sec x csc x. Using the facts that tan x =
cos x
cos x
1
1
cot x =
, sec x =
, and csc x =
, we obtain that
sin x
cos x
sin x
sin x
cos x
1
1
+
=
·
cos x
sin x
cos x sin x
sin2 x + cos2 x
1
=
cos x sin x
cos x sin x
1
1
=
cos x sin x
cos x sin x
The last line results from applying the Pythagorean Identity. The last equality is clearly true and
so the statement is proved. □
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(b) Statement: All perfect squares are positive.
Proof: Let n be an integer.
Case 1: n is positive. If n is positive, then n2 is positive because it is the product of two positive
integers.
Case 2: n is negative. If n is negative, then n2 is positive because it is the product of two negative
integers. □
3.2 Prove the following statements.
(a) For all a ∈ Z, a3 − 6a2 + 5a + 1 is odd.
(b) For all x ∈ R, 1 + 99 sin2 x ≥ 10 sin 2x.
(c) For all a, b, c ∈ Z, if a | b and a | c, then a | (b + c)2
(d) For all a ∈ N, if for all b ∈ Z, a | (6b + 8), then a = 1 or a = 2.
3.3 Let a, b, c ∈ Z. Consider the implication S(a, b, c): If a ∤ 3b and a | (b + 12c), then a ∤ 9c.
(a) State the negation of S(a, b, c). Do not use the word “not” or the negation symbol.
(b) State the converse of S(a, b, c).
(c) State the contrapositive of S(a, b, c).
(d) Prove ∀a, b, c ∈ Z, S(a, b, c).
3.4 Prove that there exists an integer k such that for every natural number x,
x3k+15 + x2 − kx + 5
is a composite integer.
3.5 Let a and b be natural numbers, not both 1. Prove that if a4 + 4b is prime, then a is odd and b is even.
Hint: Note that x4 + 4y 4 = [(x − y)2 + y 2 ][(x + y)2 + y 2 ].
3.6 Prove that no natural number a exists such that a2 − 10 is a perfect square.
3.7 For all integers w, x, y and z with w ̸= y and wz − xy ̸= 0, prove that there exists a unique rational
wr + x
= 1.
number r such that
yr + z
4
Induction
4.1 Prove that for all n ∈ N,
n
X
(−1)i i2 =
i=1
(−1)n n(n + 1)
.
2
4.2 A sequence a0 , a1 , a2 , . . . is defined by a0 = 7, a1 = 26 and am = 7am−1 − 10am−2 , for all integers m ≥ 2.
Prove that an = 3 · 2n + 4 · 5n , for all integers n ≥ 0.
4.3 Let c ∈ R such that c ̸= 0. Prove that for all n ∈ N, if c +
5
1
1
∈ Z, then cn + n ∈ Z.
c
c
Sets
5.1 Let A = {x ∈ Z : 6 | x}, B = {y ∈ Z : 9 | y} and C = {3k : k ∈ Z}. Prove that A ∪ B ⊊ C.
3
6
Greatest Common Divisor and Prime factorization
We will do this properly in class. It is still helpful to learn it here yourself first.
7
Linear Diophantine Equation
We will skip this. Useful to learn.
8
Congruences
We will speedrun this in one class. It is very helpful to learn it here yourself first.
9
RSA
We will do this in class. Safe to skip here.
10
Complex numbers
10.1 Let z = −8 + 8i, u = 3 − 5i and w = 3 + 4i. Express the following in standard form.
(a) 2z + uw − i69
√
z
− (1 − 3i)9
(b)
|w|u
10.2 Shade the region of the complex plane defined by {z + 2 : 1 < |z| < 3, z ∈ C}.
10.3 Let u, w ∈ C. Prove that if |u + 2w|2 = |2u + w|2 , then |u|2 = |w|2 .
√
10.4 Find all z ∈ C such that (z + 3 + i)6 + 64 = 0. Give your answers in standard form.
11
Polynomials
11.1 One root of the polynomial f (x) = 3x5 − 14x4 + 14x3 + 32x2 − 69x + 30 over C is 2 + i.
(a) Write f (x) as a product of irreducible polynomials in Q[x].
(b) Write f (x) as a product of irreducible polynomials in R[x].
(c) Write f (x) as a product of irreducible polynomials in C[x].
11.2 Suppose that 1 + 3i is a root of x3 + px2 + q where p and q are real. Determine the values of p and q.
11.3 Let n ∈ Z and n > 2. Find all the complex solutions to
z = z n−1 .
11.4 Prove that there does not exist a polynomial f (x) ∈ C[x] such that z = f (z) for all z ∈ C.
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