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Mathematics LibreTexts

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

  • Page ID
    7053
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    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 \(0 \le r < n\).