5.4: Summary

Last updated
Save as PDF

Page ID: 62294

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

A valid deduction (or “result”) is usually called a theorem, proposition, corollary, or lemma.
Divisibility and congruence
- Important definitions:
  - divisor, multiple
  - congruent modulo \(n: a \equiv b(\bmod n)\)
  - remainder
  - irrational number
- Congruence \((\bmod n)\) is reflexive, symmetric, and transitive
- \(a \equiv b (\bmod n)\) iff \(a\) and \(b\) have the same remainder when divided by \(n\)
- \(\sqrt{2}\) is irrational
- Notation:
  - \(a \mid b, a \nmid b\)
  - \(a \equiv b (\bmod n)\)
Commutative groups
- Important definitions:
  - commutative group (commutative, associative, identity element, negatives)
  - subgroup (closed under negatives and addition)
- The identity element of a group is unique.
- The negative of each element of a group is unique.
- Notation:
  - 0 (identity element)
  - \(−g\) (negative)
Convergent sequences
- Important definitions:
  - absolute value
  - converges
- triangle inequality
- Notation:
  - \(|x|\)
  - \(a_{n} \rightarrow L\)