Volume 0

Real Numbers and Set Notation • Real Number Set, ℝ=(−∞,∞) “the real number line”

by admin|Published May 4, 2026|2 comments

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Core Mathematical Sets and Notation

Real Number Set

$\mathbb{R}=(-\infty,\infty)\quad\text{the real number line}$

Example

$f(x)=x$

$D=\mathbb{R}=(-\infty,\infty)$

$\{x\,|\,x\in\mathbb{R}\}$

$-\infty<x<\infty$

Two Dimensional Real Space

$\mathbb{R}^2=(-\infty,\infty)\times(-\infty,\infty)$

xy-plane

Restricted Region in the xy-plane

$\mathbb{R}^2=(0,3)\times(0,2)$

This may look this way in Calculus 2 for integration:

$R:\;0\leq x\leq 3,\quad 0\leq y\leq 2$

$\int_0^3\int_0^2 f(x)\,dy\,dx$

Three Dimensional Real Space

$\mathbb{R}^3$

is in the x, y, z coordinate system

Complex Number Set

$\mathbb{C}=\{a+ib\,|\,a,b\in\mathbb{R},\;i=\sqrt{-1}\}$

All Integers

$\mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$

Positive Integers, natural numbers

$\mathbb{Z}^{+}=\mathbb{N}=\{1,2,3,\ldots\}$

Rational Numbers

$\mathbb{Q}=\left\{\frac{a}{b}\;|\;a,b\in\mathbb{Z},\;b\neq0\right\}$

Irrational Set

$\mathbb{R}\setminus\mathbb{Q}=\{e,\pi,\sqrt{2},\ldots\}$

Universal Quantifiers

In, $\in$
Not in, $\notin$
Such that, $\exists$
Therefore, $\therefore$
Because, $\because$
Exist, $\exists$
For all, $\forall$
Divides, |

Does not divide, $\nmid$
Implies, $\Rightarrow$
If and only if, $\Leftrightarrow$
Modulo or Equivalent, $\equiv$
Goes to, $\to$
Subset, $\subset$
Tilda, $\sim$
Infinity, $\infty$

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admin

5 hours ago

For the real number set, you should have it as a subset….

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admin

5 hours ago

Universal Quantifier is not technically right and it should say exists not exist ,… (samples of finding typos to get credit in the books)

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