Sets of Numbers

Sets

Symbol	Description
$N$	Set of all natural numbers ${0, 1, 2, ...}$
$Z$	Set of all integers
$Q$	Set of all rational numbers (Numbers that can be expressed as a fraction)
$R$	Set of all real numbers (Any number)
$C$	Set of all complex numbers

Subscript / Superscript

Symbol	Description
$Z^{+}$	Set of all positive integers
$R^{-}$	Set of all negative real numbers
$Z_{\geq 12}$	Set of all integers greater or equal to 12
Note: 0 is neither positive nor negative

Summation and Product

Summation

$\sum_{k = m}^{n} a_{k} = a_{m} + a_{m + 1} + ... + a_{n}$

Summation of arithmetic sequence

For arithmetic sequence $a = a_{k - 1} + d$ : $a_{n} = a_{0} + d_{n}$ $\sum_{k = 0}^{n - 1} a_{k} = \frac{n}{2} (2 a_{0} + (n - 1) d)$

Summation of geometric sequence

For geometric sequence $a_{k} = r a_{k - 1}$ : $a_{n} = a_{0} r^{n}$ $\sum_{k = 0}^{n - 1} a_{k} = a_{0} (\frac{1 - r ^{n}}{1 - r})$

Products

$\prod_{k = m}^{n} a_{k} = a_{m} \cdot a_{m + 1} \cdot ... \cdot a_{n}$

Properties (Theorem 5.1.1)

$\sum_{k = m}^{n} a_{k} + \sum_{k = m}^{n} b_{k} = \sum_{k = m}^{n} (a_{k} + b_{k})$ $c \cdot \sum_{k = m}^{n} a_{k} = \sum_{k = m}^{n} (c \cdot a_{k})$ $\prod_{k = m}^{n} (a_{k} \cdot b_{k}) = (\prod_{k = m}^{n} a_{k}) \cdot (\prod_{k = m}^{n} b_{k})$

Properties of Integers

$\forall x, y, z \in Z$ :

Closure

Integers are closed under addition and multiplication When you add or multiply an integer, you will get an integer $x + y \in Z$ and $x y \in Z$

Commutativity

$x + y = y + x$ and $x y = y x$

Associativity

$x + y + z = (x + y) + z = x + (y + z)$ and $x yz = (x y) z = x (yz)$

Distributivity

Multiplication is distributive over addition (but not the other way around) $x (y + z) = x y + x z$

Trichotomy

Exactly one of the following is true: $x = y$ or $x < y$ or $x > y$

🪴 Notes Hehe

Explorer

General