Mathematical Symbols and Foundations

Mathematics uses a vast array of symbols to express numbers, operations, relations, and abstract concepts. Here is a complete glossary of the most commonly used math symbols along with their meanings and examples.

Author

Benedict Thekkel

Symbols

1. Basic Arithmetic Symbols

Symbol	Name	Meaning	Example
`+`	Plus	Addition	$5 + 3 = 8$
`-`	Minus	Subtraction	$10 - 4 = 6$
`\times$ or`*`\| Multiply \| Multiplication \| $4 \times 3 = 12$ \| \|`$ or `/`	Divide	Division	$12 \div 4 = 3$
`=`	Equals	Equality	$7 + 2 = 9$
`\neq$ \| Not equal \| Inequality \| $5 \neq 3$ \| \|`<$	Less than	Comparison	$2 < 5$
`>$ \| Greater than \| Comparison \| $8 > 6$ \| \|`$	Less than or equal	Comparison	$4 \leq 4$
`$	Greater than or equal	Comparison	$10 \geq 7$

2. Algebra Symbols

Symbol	Name	Meaning	Example
$x, y, z$	Variables	Unknown values	$x + 2 = 5$
$a^{n}$	Power	Exponentiation	$2^{3} = 8$
$\sqrt{x}$	Square Root	Root of a number	$\sqrt{9} = 3$
$\sqrt[3]{x}$	Cube Root	Cube root of a number	$\sqrt[3]{27} = 3$
$\| x \|$	Absolute Value	Distance from zero	$\| - 5 \| = 5$
$\sum$	Summation	Sum of a sequence	$\sum_{i = 1}^{n} i = 1 + 2 + . . . + n$
$\prod$	Product	Product of a sequence	$\prod_{i = 1}^{n} i = 1 \cdot 2 \cdot . . . \cdot n$
$\approx$	Approximately equal	Approximate value	$π \approx 3.14$
$\equiv$	Identical to	Congruent or equivalent	$a \equiv b \mod n$
$\Rightarrow$	Implies	Logical implication	$x > 2 \Rightarrow x^{2} > 4$
$\Leftrightarrow$	If and only if	Logical equivalence	$x^{2} = 4 \Leftrightarrow x = \pm 2$

3. Set Theory Symbols

Symbol	Name	Meaning	Example
${}$	Set	Collection of elements	$A = {1, 2, 3}$
$\in$	Element of	Belongs to a set	$2 \in A$
$\notin$	Not an element of	Not in a set	$4 \notin A$
$\emptyset$	Empty Set	No elements	$B = \emptyset$
$\cup$	Union	Elements in either set	$A \cup B$
$\cap$	Intersection	Common elements in sets	$A \cap B$
$\subseteq$	Subset	All elements in another set	$A \subseteq B$
$\subset$	Proper Subset	Strictly contained	$A \subset B$
$\supseteq$	Superset	Contains all elements	$B \supseteq A$
$∖$	Set Difference	Elements in one but not both	$A ∖ B$
$\times$	Cartesian Product	Ordered pairs of elements	$A \times B$
$\| A \|$	Cardinality	Number of elements	$\| A \| = 3$

4. Logic Symbols

Symbol	Name	Meaning	Example
$\neg$	Not	Negation	$\neg P$
$\land$	And	Conjunction	$P \land Q$
$\lor$	Or	Disjunction	$P \lor Q$
$\Rightarrow$	Implies	Conditional statement	$P \Rightarrow Q$
$\Leftrightarrow$	If and only if	Biconditional	$P \Leftrightarrow Q$
$\forall$	For all	Universal quantifier	$\forall x, x > 0$
$\exists$	There exists	Existential quantifier	$\exists x, x^{2} = 4$
$\exists!$	There exists exactly one	Unique existence	$\exists! x, x^{2} = 1$

5. Calculus Symbols

Symbol	Name	Meaning	Example
$\frac{d}{d x}$	Derivative	Rate of change	$\frac{d}{d x} x^{2} = 2 x$
$\int$	Integral	Area under curve	$\int x d x = \frac{x^{2}}{2} + C$
$\nabla$	Nabla (Gradient)	Vector derivative	$\nabla f (x, y)$
$\partial$	Partial Derivative	Derivative of multivariable	$\frac{\partial f}{\partial x}$
$\infty$	Infinity	Unbounded value	$lim_{x \to \infty} \frac{1}{x} = 0$
$lim$	Limit	Approaching a value	$lim_{x \to 0} \frac{\sin (x)}{x} = 1$

6. Probability & Statistics Symbols

Symbol	Name	Meaning	Example
$P (A)$	Probability	Likelihood of event A	$P (A) = 0.5$
$E (X)$	Expected Value	Mean of random variable X	$E (X) = μ$
$Var (X)$	Variance	Spread of random variable X	$Var (X) = σ^{2}$
$σ$	Standard Deviation	Square root of variance	$σ = \sqrt{Var (X)}$
$\sum$	Summation	Sum of values	$\sum X_{i}$
$n!$	Factorial	Product of first n integers	$4! = 4 \times 3 \times 2 \times 1 = 24$
$(\binom{n}{k})$	Combination	Ways to choose k from n	$(\binom{5}{2}) = 10$
$P (n, k)$	Permutation	Ordered arrangements	$P (5, 2) = 20$

7. Complex Numbers & Linear Algebra

Symbol	Name	Meaning	Example
$i$	Imaginary Unit	$\sqrt{- 1}$	$i^{2} = - 1$
$\| v \|$	Vector Magnitude	Length of vector	$\| v \| = \sqrt{x^{2} + y^{2}}$
$\cdot$	Dot Product	Scalar product of vectors	$u \cdot v = u_{x} v_{x} + u_{y} v_{y}$
$\times$	Cross Product	Vector product	$u \times v$
$A^{T}$	Transpose	Flip matrix dimensions	$A^{T}$

Summary of Mathematical Axioms in Table Format

Mathematical axioms are fundamental truths that serve as the building blocks for all mathematical reasoning. Here is a comprehensive table summarizing the main axioms of mathematics along with examples.

1. Axioms of Real Numbers (Field Axioms)

These axioms define the properties of real numbers under addition and multiplication.

Axiom	Definition	Example
1. Closure	Sum/product of two real numbers is real	$ a + b $ is real, $ ab $ is real
2. Commutativity	Order doesn’t affect sum/product	$ a + b = b + a $, $ ab = ba $
3. Associativity	Grouping doesn’t affect sum/product	$ (a + b) + c = a + (b + c) $
4. Identity	Exists an identity element for addition and multiplication	$ a + 0 = a $, $ a = a $
5. Inverse	Additive and multiplicative inverses exist	$ a + (-a) = 0 $, $ a = 1 $
6. Distributivity	Multiplication distributes over addition	$ a(b+c) = ab + ac $

2. Axioms of Equality

These define the properties of equality used in equations and proofs.

Axiom	Definition	Example
1. Reflexive	Anything is equal to itself	$ a = a $
2. Symmetric	If $ a = b $, then $ b = a $	If $ x = y $, then $ y = x $
3. Transitive	If $ a = b $ and $ b = c $, then $ a = c $	If $ x = y $ and $ y = z $, then $ x = z $
4. Substitution	If $ a = b $, then $ a $ can be substituted for $ b $ in any expression	If $ x = 5 $, then $ x + 2 = 7 $
5. Addition Property	Adding same number to both sides preserves equality	If $ x = y $, then $ x + c = y + c $
6. Multiplication Property	Multiplying both sides by same number preserves equality	If $ x = y $, then $ cx = cy $

3. Axioms of Order

These define the properties of inequalities among real numbers.

Axiom	Definition	Example
1. Trichotomy	Exactly one is true: $ a < b $, $ a = b $, or $ a > b $	$ 3 < 4 $, $ 4 > 3 $, $ 3 = 3 $
2. Transitivity	If $ a < b $ and $ b < c $, then $ a < c $	If $ 2 < 4 $ and $ 4 < 6 $, then $ 2 < 6 $
3. Addition Property	Adding same number to both sides of inequality maintains order	If $ a < b $, then $ a+c < b+c $
4. Multiplication Property	Multiplying both sides by positive preserves order; negative reverses order	$ 2 < 3 < 3 $
5. Density	Between any two real numbers, there’s another real number	Between 1 and 2, there’s 1.5

4. Peano’s Axioms (Axioms of Natural Numbers)

Peano’s axioms define the properties of natural numbers (0, 1, 2, …).

Axiom	Definition	Example
1. Zero Axiom	0 is a natural number	$ 0 $
2. Successor Axiom	Every natural number has a successor	$ S(0) = 1, S(1) = 2 $
3. Distinctness	No two numbers have the same successor	$ S(n) S(m) $ if $ n m $
4. Zero is Not Successor	0 is not the successor of any number	$ n, S(n) = 0 $
5. Induction	If 0 has property $ P $, and $ n $ having $ P $ implies $ S(n) $ has $ P $, then all natural numbers have $ P $	Inductive proofs

5. Axioms of Set Theory (Zermelo-Fraenkel Axioms)

These form the foundation of modern set theory.

Axiom	Definition	Example
1. Axiom of Extensionality	Two sets are equal if they have the same elements	$ A = B $ if $ x (x A x B) $
2. Axiom of Empty Set	There exists a set with no elements	$ = {} $
3. Axiom of Pairing	For any two sets, there exists a set containing exactly those two	$ {a, b} $
4. Axiom of Union	For any set, there exists a union set containing all elements of its subsets	$ A B $
5. Axiom of Power Set	For any set, there exists a set of all its subsets	$ P(A) $
6. Axiom of Infinity	There exists an infinite set	$ = {0, 1, 2, …} $
7. Axiom of Replacement	Elements can be replaced by other elements based on a rule	$ f: A B $
8. Axiom of Choice	From any set of non-empty sets, one element can be chosen from each	Well-ordering theorem

6. Euclidean Geometry Axioms (Euclid’s Postulates)

These are the foundations of Euclidean geometry.

Axiom	Definition	Example
1. Line Postulate	A line can be drawn through any two points	Through $ A $ and $ B $, draw $ AB $
2. Line Segment Extension	Any line segment can be extended indefinitely	$ AB A’B’ $
3. Circle Postulate	A circle can be drawn with any center and radius	Circle with center $ O $ and radius $ r $
4. Right Angle Postulate	All right angles are congruent	$ A = 90^, B = 90^A B $
5. Parallel Postulate	If a line intersects two lines and the interior angles sum to less than 180°, the two lines will intersect	Basis of Euclidean geometry

Major Topics in Mathematics

Mathematics is a vast field with numerous branches, each with its own set of concepts, theories, and applications. Here is a comprehensive overview of the major topics in mathematics.

1. Arithmetic

Study of numbers and basic operations.

Subtopic	Description	Example
Natural Numbers	Counting numbers (1, 2, 3, …)	$ = {1, 2, 3, …} $
Integers	Positive, negative, and zero	$ = {…,-2, -1, 0, 1, 2,…} $
Fractions	Division of integers	$ , $
Decimals	Fractional numbers in decimal form	$ 0.75, 3.14 $
Order of Operations	Sequence for operations (PEMDAS)	$ 2 + 3 = 14 $
Prime Numbers	Numbers with only two factors (1, itself)	$ 2, 3, 5, 7, 11 $
Divisibility Rules	Rules to check if one number divides another	$ 4 $

2. Algebra

Study of mathematical symbols and rules for manipulating them.

Subtopic	Description	Example
Variables and Expressions	Symbols representing numbers	$ x + 2, 3y - 7 $
Equations	Mathematical statements of equality	$ x + 3 = 7 $
Inequalities	Mathematical statements of inequality	$ x > 5, y $
Polynomials	Expressions with multiple terms	$ x^2 + 3x + 2 $
Factoring	Breaking down expressions into factors	$ x^2 - 4 = (x-2)(x+2) $
Functions	Relations between inputs and outputs	$ f(x) = x^2 + 2x + 1 $
Linear Algebra	Study of vectors, matrices, and systems of equations	$ Ax = b $
Quadratic Equations	Equations of the form $ ax^2 + bx + c = 0 $	$ x^2 - 5x + 6 = 0 $

3. Geometry

Study of shapes, sizes, and properties of space.

Subtopic	Description	Example
Euclidean Geometry	Study of flat surfaces and shapes	Points, lines, angles, triangles
Coordinate Geometry	Geometry using the Cartesian plane	Distance, midpoint, slope
Solid Geometry	Study of 3D shapes	Cubes, spheres, cones
Trigonometry	Study of angles and lengths in triangles	$ , , $
Congruence and Similarity	Comparisons of shapes	$ ABC DEF $, $ PQR XYZ $
Transformations	Translations, rotations, reflections, scaling	$ T(x, y) = (x+2, y-3) $
Area and Volume	Measure of space inside shapes	Area of circle: $ r^2 $

4. Calculus

Study of change and motion.

Subtopic	Description	Example
Limits	Behavior of functions as inputs approach a value	$ _{x } = 1 $
Derivatives	Rate of change of a function	$ f’(x) = 2x $ for $ f(x) = x^2 $
Integration	Accumulation of quantities	$ x dx = + C $
Differential Equations	Equations involving derivatives	$ y’ + y = 0 $
Multivariable Calculus	Calculus with more than one variable	$ f(x, y) = x^2 + y^2 $
Vector Calculus	Calculus of vector fields	$ F $, $ F $
Series and Sequences	Summation of sequences	$ _{n=1}^{} $

5. Probability and Statistics

Study of randomness, data analysis, and interpretation.

Subtopic	Description	Example
Probability Theory	Study of random events and likelihoods	$ P(A) = $
Random Variables	Variables representing outcomes	$ X: $
Distributions	Patterns of data spread	Normal, Binomial, Poisson
Descriptive Statistics	Measures of central tendency and spread	Mean, Median, Mode, Variance
Inferential Statistics	Drawing conclusions from data samples	Hypothesis testing, Confidence Intervals
Regression Analysis	Relationship between variables	Linear Regression: $ y = mx + c $
Correlation	Degree of relationship between variables	$ -1 r $

6. Number Theory

Study of integers and their properties.

Subtopic	Description	Example
Prime Numbers	Numbers divisible only by 1 and itself	$ 2, 3, 5, 7, 11 $
Divisibility	Rules for determining divisibility	$ 4 $
GCD and LCM	Greatest Common Divisor, Least Common Multiple	$ (12, 15) = 3 $
Modular Arithmetic	Remainders in division	$ 7 $
Diophantine Equations	Equations with integer solutions	$ x^2 + y^2 = z^2 $
Fermat’s Last Theorem	$ x^n + y^n z^n $ for $ n > 2 $	Proved by Andrew Wiles

7. Discrete Mathematics

Study of countable structures and logic.

Subtopic	Description	Example
Logic and Propositions	Boolean logic, truth values	$ P Q, P, P Q $
Set Theory	Study of collections of objects	$ A = {1, 2, 3} $
Combinatorics	Counting and arrangement techniques	Permutations, Combinations
Graph Theory	Study of graphs and networks	Nodes, Edges, Eulerian Path
Boolean Algebra	Algebra of true/false values	$ A B, A B, A $
Algorithms and Complexity	Study of problem-solving methods and efficiency	Sorting, Searching Algorithms

8. Applied Mathematics

Mathematics used in practical fields.

Subtopic	Description	Example
Mathematical Physics	Equations describing physical phenomena	Maxwell’s Equations, Schrödinger’s Equation
Engineering Mathematics	Math in engineering applications	Control systems, Signal processing
Financial Mathematics	Math for finance and investment	Compound Interest, Derivatives
Cryptography	Secure communication using mathematics	RSA Algorithm, Hash Functions
Game Theory	Strategic decision-making	Nash Equilibrium