Mathematical Symbols and Their Meanings

Basic Arithmetic Symbols

• Plus sign (+): Represents addition
• Minus sign (-): Represents subtraction; also denotes negative numbers
• Multiplication sign (× or ·): Used to denote multiplication
• Division sign (÷ or /): Signifies division
• Plus-minus sign (±): Denotes either plus or minus; sometimes indicates a range of values
• Minus-plus sign (∓): Used with ± to denote the opposite of ± (e.g., if ± is used for a+b, then ∓ can be used for a-b)

Root and Equality Symbols

• Root symbol (√): Denotes the square root of a number
  • N root (ⁿ√): Represents the Nth root of a number
• Equal sign (=): Indicates equality between two expressions
• Not equal sign (≠): Indicates that two expressions are not equal

Approximation Symbols

• Approximately equal sign (≈): Shows that two values are close but not exactly equal
• Tilde (~): Can denote approximation, similarity, or proportionality
• Proportionality sign (∝): Represents proportionality
• Triple bar (≡): Indicates congruence in modular arithmetic or identity

Comparison Symbols

• Less than sign (<): Indicates one quantity is smaller than another
• Greater than sign (>): Indicates one quantity is larger than another
• Less than or equal to (≤): Indicates one value is smaller than or equal to another
• Greater than or equal to (≥): Indicates one value is larger than or equal to another
• Much less than (≪): Indicates one quantity is much smaller than another
• Much greater than (≫): Indicates one quantity is much larger than another

Set Theory Symbols

• Empty set (∅): Denotes a set with no elements
• Number sign (#): Denotes the cardinality of a set (number of elements)
• In symbol (∈): Indicates membership in a set
• Not in symbol (∉): Indicates non-membership in a set
• Set inclusion (⊆): Represents that one set is a subset of another set
  • Strict set inclusion (⊂): Indicates that one set is a proper subset of another set
  • Subset but can be equal (⊇): Used to emphasize that sets may be equal
• Union (∪): Operation to combine two sets into one set containing unique elements
• Intersection (∩): Operation to combine two sets into one set containing common elements
• Set difference (): Represents elements of the first set not in the second set
• Symmetric difference (Δ or ⊖): Set containing elements in either one of the sets but not in both

Logical Symbols

• Negation (¬): Indicates the opposite of a statement
• OR (∨): Returns true if at least one operand is true
• AND (∧): Returns true only if both operands are true
• Exclusive OR (⊻): Returns true if exactly one operand is true
• True constant (⊤): A logical constant for true value
• False constant (⊥): A logical constant for false value
• Universal quantifier (∀): Asserts that a statement is true for all elements in a given domain
• Existential quantifier (∃): Asserts that there exists at least one element for which a statement is true
• Uniqueness quantifier (∃!): Indicates there is exactly one element for which a statement is true
• Conditional operator (→): Denotes implication (if first statement is true, then the second is true)
• Logical equivalence operator (↔): Indicates two statements have the same logical value

Number Systems

• 𝕌: Universal set
• ℕ: Natural numbers
• ℤ: Integers
• ℚ: Rational numbers
• ℝ: Real numbers
• ℂ: Complex numbers
• ℍ: Quaternions
• 𝕆: Octonions

Calculus Symbols

• Apostrophe ('): Denotes the derivative of a function (f')
  • Double apostrophe ("↻)
  • Newton's notation (˙): Derivative with respect to time
• Leibniz notation (d/dx): Derivative with the variable at the bottom
  • Partial derivative (∂/∂x): For function of multiple variables
• Integral (∫): Denotes an anti-derivative
  • Definite integral (∫_a^b): Represents the area under a curve over an interval_

Function Symbols

• Arrow (→): Used to define a function without naming it
• Function composition (∘): Operation that combines two functions

Logarithms and Limits

• Logarithm (log_b): Inverse operation of exponentiation
  • Base 10 logarithm (log)
  • Natural logarithm (ln): Base e
• Limit (lim): Denotes the behavior of a function as input approaches a certain value

Complex Numbers

• ℜ: Real part of a complex number
• ℑ: Imaginary part of a complex number
• Complex conjugate (z̅): Changes sign of the imaginary part

Series and Products

• Summation (Σ): Sum of a series of terms
• Product (Π): Product of a series of terms

Infinity and Factorials

• Infinity symbol (∞): Denotes unlimited ess
• Aleph (ℵ): Cardinality of infinite sets
  • ℵ₀: Cardinality of the set of natural numbers
• 𝔠: Cardinality of the set of real numbers
• Factorial (n!): Product of all positive integers smaller than or equal to n

Binomials and Absolute Values

• Binomial coefficient (bin): Number of ways to choose k elements from n elements
• Absolute value (|a|): Distance from zero on the number line

Floor, Ceiling, and Nearest Integer

• Floor function (⌊x⌋): Greatest integer less than or equal to x
• Ceiling function (⌈x⌉): Smallest integer greater than or equal to x
• Nearest integer function (⌊x⌉): Returns the nearest integer to a given value

Geometry and Number Properties

• Divisibility (|): Indicates divisibility
• Non-divisibility (∤): Indicates non-divisibility
• Parallelism (∥): Denotes parallel lines
• Non-parallelism (≁): Denotes non-parallel lines
• Perpendicularity (⊥): Indicates perpendicular lines; can also mean co-prime numbers
• Line segment (---): Line segment between two points
• Ray (→): Ray starting at one point and passing through another
• Infinite line (↔): Line passing through two points in both directions