1.S: Introduction to Writing Proofs in Mathematics (Summary)
- Page ID
- 7036
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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 \(a^2 + b^2 = c^2\).