1.S: Introduction to Writing Proofs in Mathematics (Summary)
 Page ID
 7036
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{\!\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
Important Definitions
 Statement
 Odd integer
 Conditional statement
 Even integer
 Pythagorean triple
Important Number Systems and Their Properties
 The natural numbers, \(\mathbb{N}\); the integers, \(\mathbb{Z}\); the rational numbers, \(\mathbb{Q}\); and the real number, \(\mathbb{R}\).
 Closure Properties of the Number Systems
Number System Closed Under Natural numbers, \(\mathbb{N}\) addition and multiplication Integers, \(\mathbb{Z}\) addition, subtraction, and multiplication Rational numbers, \(\mathbb{Q}\) addition, subtraction, and multiplication, and division by nonzero rational numbers Real number, \(\mathbb{R}\) addition, subtraction, and multiplication, and division by nonzero real numbers  Inverse,commutative,associative, and distributive properties of the real numbers.
Important Theorems and Results

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.

The Pythagorean Theorem. If \(a\) and \(b\) are the lengths of the legs of a right triangle and \(c\) is the length of the hypotenuse, then \(a^2 + b^2 = c^2\).