Notation

Foundations of mathematics

Introduction

Mathematics is an elegant language. Unless we define a consistent notation to describe mathematics, it may become cumbersome and difficult to comprehend. At The Learning Machine, we are committed to ensuring a uniform and possibly, standard notation across all learning material. In some cases, this may require us to diverge from commonly used variables and styles from popular texts and research papers, but that is the price we pay to have site-wide consistency.

Sets

For sets, we will adhere to the following notation.

Entity Notation Comments
An element of a set \( a, b, \ldots \) Lowercase alphabets
A set \( \sA, \sB, \ldots \) Uppercase block letters
Elements of set \( \sA \) \( \sA = \set{a, b, c \ldots} \) elements enclosed in curly braces
Set builder notation \( \sA = \lbrace x : \text{a property that defines which } x\text{'s are included} \rbrace \)
\( \sA \) is equal to \( \sB \) \( \sA = \sB \)
\( \sA \) is not equal to \( \sB \) \( \sA \ne \sB \)
\( \sA \) is subset of \( \sB \) \( \sA \subseteq \sB \)
\( \sA \) is a proper subset of \( \sB \) \( \sA \subset \sB \)
\( \sA \) is not a subset of \( \sB \) \( \sA \not\subset \sB \)
Cardinality of \( \sA \) \( \cardinality{\sA} \)
Complement of a set \( \sA \) \( \complement{\sA} \)
Power set of a set \( \sA \) \( \powerset{\sA} \)

Special sets

Special sets Notation
Empty set \( \emptyset \)
Natural numbers \( \natural \)
Real numbers \( \real \)
Integers (positive negative, and zero ) \( \integer \)
Rational numbers \( \rational \)
Irrational numbers \( \irrational \)
Complex numbers \( \complex \)
Open interval \( (a,b) \)
Closed interval \( [a,b] \)
Half-open interval \( [a,b) \) or \( (a,b] \)

Set operations

Operation (over sets \( \sA \) and \( \sB \)) Notation
Union \( \sA \cup \sB \)
Intersection \( \sA \cap \sB \)
Difference \( \sA - \sB \) or \( \sA \setminus \sB \)
Symmetric difference \( \sA \oplus \sB \)
Cartesian product \( \sA \times \sB \)

Sequences

Entity Notation Comments
A sequence \( \sA = \seq{a, b, \ldots} \) elements enclosed in round parentheses
An \(N\)-tuple \( \seq{a_1,a_2,\ldots,a_n} \) \( N \) elements enclosed in round parentheses
An ordered pair \( (a,b) \) elements enclosed in round parentheses

Relations and functions

For relations and functions, we will adhere to the following notation.

Entity Notation Comments
Domain of a relation \( \text{dom}(R) \)
Range of a relation \( \text{range}(R) \)
Inverse of a relation \( R^{-1} \)
Function from a set to another \( f: \sA \to \sB \)
Range of a function \( \text{range}(f) \)
Composite function \( (g \circ f)(a) = g(f(a)) \)

Logic

For mathematical logic, we will adhere to the following notation.

Entity Notation Comments
Proposition \( P, Q, R \) Capital letters
Predicate \( P(x) \) A proposition with variable
Truth values \( T \) or \( F \) For True and False respectively
Negation of a proposition \( \neg P \)
Disjunction of propositions \( P \vee Q \)
Conjunction of propositions \( P \wedge Q \)
Exclusive-OR of propositions \( P \oplus Q \)
Implication \( P \Rightarrow Q \)
Biconditional \( P \Leftrightarrow Q \)
Equivalent propositions \( P \equiv Q \)
Universal quantifier \( \forall x \in S, P(x) \) "For every \( x \in S, P(x) \)"
Existential quantifier \( \exists x \in S, P(x) \) "For some \( x \in S, P(x) \)"

Calculus

For calculus, we will adhere to the following notation.

Entity Notation Comments
Derivatives \( \frac{df}{dx} \) Leibniz's notation
Derivatives: shorthand \( f'(x) \) Lagrange's notation

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