$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

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

• 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 $$0 \le r < n$$.