3.S: Constructing and Writing Proofs in Mathematics (Summary)

Important Definitions

• Divides,divisor,page82
• Factor, multiple, page 82
• Proof, page 85
• Undefined term, page 85
• Axiom, page 85
• Definition,page86
• Conjecture, page 86
• Theorem, page 86
• Proposition,page 86
• Lemma, page 86
• Corollary, page 86
• Congruence modulo $n$, page 92
• Tautology,page 40
• Contradiction,page 40
• Absolutevalue,page 135

Important Theorems and Results about Even and Odd Integers

• Exercise (1), Section 1.2
  If $m$ is an even integer, then $m + 1$ is an odd integer.
  If $m$ is an odd integer, then $m + 1$ is an even integer.
• Exercise (2), Section 1.2
  If $x$ is an even integer and $y$ is an even integer, then $x + y$ is an even integer.
  If $x$ is an even integer and $y$ is an odd integer, then $x + y$ is an odd integer.
  If $x$ is an odd integer and $y$ is an odd integer, then $x + y$ is an even integer.
• Exercise (3), Section 1.2. If $x$ is an even integer and $y$ is an integer, then $x \cdot y$ is an even integer.
• Theorem1.8. If $x$ is an odd integer and $y$ is an odd integer, then $x \cdot y$ is an odd integer.
• Theorem 3.7. The integer $n$ is an even integer if and only if $n^2$ is an even integer.
  Preview Activity $\PageIndex{2}$ in Section 3.2. The integer $n$ is an odd integer if and only if $n^2$ is an odd integer.

Important Theorems and Results about Divisors

• Theorem 3.1. For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a | b$ and $b | c$, then $a | c$.
• Exercise (3), Section 3.1. For all integers $a$, $b$, and $c$ with $a \ne 0$,
  If $a | b$ and $a | c$, then $a | (b + c)$.
  If $a | b$ and $a | c$, then $a | (b - c)$.
• Exercise (3a), Section 3.1. For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a | b$, then $a | (bc)$.
• Exercise (4), Section 3.1. For all nonzero integers $a$ and $b$, if $a | b$ and $b | a$, then $a = \pm b$.

The Division Algorithm

Let $a$ and $b$ be integers with $b > 0$. Then there exist unique integers $q$ and $r$ such that

$a = bq + r$ and $0 \le r < b$.

Important Theorems and Results about Congruence

• Theorem 3.28. Let $a, b, c \in \mathbb{Z}$ and let $n \in \mathbb{N}$. If $a \equiv b$ (mod $n$) and $c \equiv d$ (mod $n$), then
  
  $(a + c) \equiv (b + d)$ (mod $n$).
  
  $ac \equiv bd$ (mod $n$).
  For each $m \in \mathbb{N}$, $a^m \equiv b^m$ (mod $n$).
• Theorem 3.30. For all integers a, b, and c,
  Reflexive Property. $a \equiv a$ (mod $n$).
  Symmetric Property. If $a \equiv b$ (mod $n$), then $b \equiv a$ (mod $n$).
  Transitive Property. If $a \equiv b$ (mod $n$) and $b \equiv c$ (mod $n$), then $a \equiv c$ (mod $n$).
• Theorem 3.31. Let $a \in \mathbb{Z}$ and let $n \in \mathbb{N}$. If $a = nq + r$ and $0 \le r < n$ for some integers $q$ and $r$, then $a \equiv r$ (mod $n$).
• Corollary 3.32. Each integer is congruent, modulo $n$, to precisely one of the integers 0, 1, 2, ..., $n - 1$. That is, for each integer $a$, there exists a unique integer $r$ such that

$a \equiv r$ (mod $n$) and $0 \le r < n$.