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Mathematics LibreTexts

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 \(a^2 + b^2 = c^2\).