Notation

Sets

Sets	Meaning
$N$	Natural numbers
$Z$	Integers
$Q$	Rational numbers
$R$	Real numbers
$C$	Complex numbers
$H$	Quaternions
$P$	Prime numbers
$\emptyset$	Empty Set

Notation

Notation	Meaning
$a \in G$	$a$ in $G$
$\forall a$	for all values of $a$
$\exists a$	there exists $a$
$\exists! a$	there exists unique $a$
$\sum_{i = a}^{b} i$	summation of i from a to b
$⋃_{i = a}^{b} S_{i}$	the union of all the sets $S_{a}$ to $S_{b}$
$lim_{x \to a} f (x)$	the value $f (x)$ approaches as $x$ approaches $a$
$sup_{x \in [a, b]} f (x)$	the smallest upper bound of $f (x)$ over $x \in [a, b]$
$max_{x \in [a, b]} f (x)$	the maximum of $f (x)$ over $x \in [a, b]$
$∴$	Therefore
$∵$	Because
$x \mapsto y$	Elements from $x$ maps to $y$
$f : S \to T$	function maps elements from $S$ to $T$
$a ⟹ b$	$a$ implies $b$ (if $a$ is true, $b$ is true)
$a ⟺ b$	$a$ is true if and only if $b$ is true

Logical Operators

Logic Operators	Meaning
$\neg a$	NOT a
$a \land b$	a AND b
$a \lor b$	a OR b
$a \to b$	IF a THEN b (a implies b)

On the distinction between

s u p

and

m a x

In terms of sets, the maximum is the largest member of the set, while the supremum is the smallest upper bound of the set.

So, consider $A = {1, 2, 3, 4}$ . Assuming we're operating with the normal reals, the maximum is 4, as that is the largest element. The supremum is also 4, as four is the smallest upper bound.

However, consider the set $B = {x \in R | x < 2}$ . Then, the maximum of B is not 2, as 2 is not a member of the set; in fact, the maximum is not well defined. The supremum, though is well defined: 2 is clearly the smallest upper bound for the set.

References

https://math.libretexts.org/Courses/Mount_Royal_University/Mathematical_Reasoning/1%3A_Basic_Language_of_Mathematics/1.1%3A_Compound_Statements#Example_.5C(.5CPageIndex.7B1.7D.5C):