# FOM Assignment 4

Problem 1.  Exercises 8 and 10 from Section 2.5

Problem 2.  What are “DeMorgan’s Laws?”

Problem 3.  Exercises 9 and 10 from Section 2.6

Problem 4.  Exercises 4, 5, and 8 from Section 2.7

Problem 5.  Exercises 4, 10, and 11 from Section 2.10

Problem 6.  Describe at least two facts that our textbook explicitly claims we will accept without proof or justification.

Problem 7.  Let$a, b, c \in \mathbb{N}$.  Prove that $\text{lcm}(ca, cb) = c\text{lcm}(a, b)$.  (Be sure you are only using the definition of least common multiple.)

Problem 8.  Here is a proof of some statement.  Read it, and then write down the claim that was proved.

proof.  Let $n \in \mathbb{Z}$ be  arbitrary.  If $\exists a \in \mathbb{Z}$ so that $n = 2a$, then $n$ is even.  Suppose, then, that $n$ is not even.  This means that for all integers $x \in \mathbb{Z}, n \neq 2x$.  By the Division Algorithm, there exists $q, r \in \mathbb{Z}$ so that $n = 2q + r$ where $0 \leq r < 2$.  This implies that either $r = 0$ or $r = 1$.  However, if $r = 0$, then we have $n = 2q$, which is impossible since we have assumed $n$ is not even.  Therefore $r = 1$ and we have that $n = 2q + 1$.  This implies that $n$ is odd, completing the proof.  $\square$.

Problem 9.  Prove that if $a\mid b$ and if $a\mid c$ then $a\mid(b+c)$.

Problem 10.  Prove that if $a^2 \mid a$, then $a \in \{-1, 0, 1\}$.

Problem 11.  Exercises 4, 8, 14, and 26 from Chapter 4 exercises.