Homework 3 (due Thursday 2/11): Problems 1.22, 1.23,
Show that a | b and b | a implies \( a = \pm b \).
Let \( a > 1 \) be an integer which is not prime. By using contradiction, prove that it has a prime divisor p such that
\( p \leq \sqrt{a} \).
Homework 4 and Presentations I (due Thursday 2/18): Problem 1.19 (presenter: Alexandre), Problem 1.28 (presenter: Bull), Problem 1.29 (presenter: Comerford), Prove by induction \( \sum_{i=1}^n \frac{1}{i(i+1)}=\frac{n}{n+1} \) (presenter: Coon).
Homework 5 and Presentations II (due Thursday 2/25): Problem 2.3 (Dobles), Problem 2.4 (Esernio), Write a proof that for every integer x, x is even if and only if x + 1 is odd (Fank), Problem 2.5 (Galerneau), Problem 2.7(a)-(f)(Habib)
Homework 6 and Presentations III (due Thursday 3/3): Problem 2.9 (Kelly) , Problem 2.10 (Lagussi), Problem 2.14 (Martin), Problem 2.15 (Morrison) and Problem 2.16 (Patricola).
Homework 7 and Presentations IV (due Tuesday 3/29): Problem 3.1 (Purdy), Problem 3.2 (Reid), Problem 3.3 (Sperber), Problem 3.4 (a),(b) (Stavdal), Problem 3.4 (c),(d) (Suguitan), Problem 3.5 (a),(b) (Tano), Problem 3.5 (c),(d) (Varady).
Homework 8 and Presentations V (due Thursday 4/7):
Problem 3.9: (a),(b) (Syed)
Problem 3.9 (c) and a geometric explanation for eq. classes in parts (a),(b) (Vega)
Problem 3.11 (a) and Problem 3.12 (a) (Alexandre)
Problem 3.11 (b) and Problem 3.12 (b) (Bull)
Problem 3.11 (c) and Problem 3.12 (c) (Coon)
Problem 3.15 (a) (Dobies)
Problem 3.15 (b) (Esernio)
Homework 9 and Presentations VI (due Thursday 4/14)
1. Problem 3.18 (a) Problem 3.19 (a) (Fank)
2. Problem 3.18 (b) and Problem 3.19 (b) (Galerneau)
3. Problem 3.18 (c) and Problem 3.19 (c) (Habib)
4. Let \( f : X \rightarrow Y \) and \( A,B \subset X \). Show that \( f(A) \setminus f(B) \subset f(A \setminus B) \). Show that
if \( f(A \setminus B) \cap f(B)=\emptyset \) then \(f(A \setminus B)=f(A) \setminus f(B) \) (Quinn)
5. Show that \( f : X \rightarrow Y \) is injective iff \( f^{-1}(f(A))=A \) for all subsets \(A \subset X \) (Lagussi).
Homework 10 and Presentations VII (due Thursday 4/21)
1.(By using PP) show that in any group of six people there are either three mutual friends or three mutual strangers (Martin).
2. For a real number x, let \( ||x|| \) denote the distance from x to its closest integer. (By using PP) prove that for any irrational number \( \alpha \) and any
\( N \in \mathbb{N} \), there exists \( m \in \mathbb{N} \) such that
\( 0 < ||m \alpha || \leq \frac{1}{N} \). (Morrison)
3. Let \( P \) be a square with sides of length one. Fifty-one (51) points are randomly chosen inside the square. (By using PP) show that there are at least three points whose mutual distances are \( \leq \sqrt{0.08} \) (Comerford)
4. Problem 3.23 (Patricola)
5. Show that \( \mathbb{N} \) is equipotent to \( \mathbb{N} \times \mathbb{N} \) by using Theorem 3.8.(Purdy)
6. Show that if \( X \) is equipotent to \( X' \) and \( Y \) is equipotent to \( Y' \) then \( X \times Y \) is equipotent to \( X ' \times Y' \). (Reid)
Homework 11 and Presentations VIII (due Thursday 4/29)
1. Problem 4.2 (part 3) (Sperber)
2. Problem 4.2 (part 4) (Stavdal)
3. Problem 4.3 (Syed)
4. Problem 4.10 (Tano)
5. Problem 4.12 (Varady)
6. Problem 4.13 (Vega)
7. Problem 4.15 (Ventura)