Mathematical Symbols and Their Meanings

Basic Arithmetic Symbols

• Plus Sign (+): Symbol for addition.
• Minus Sign (-): Represents subtraction and negative numbers.
• Multiplication Sign (×): Denotes multiplication. Can also be represented as a dot (•).
• Division Sign (÷): Signifies division, can be written as a slash (/).
• Plus-Minus Sign (±): Indicates either plus or minus, can denote a range of values.
• Minus-Plus Sign (∓): Represents the opposite sign of plus-minus.

Roots and Powers

• Root Symbol (√): Denotes square root; can have an integer superscript for Nth root.
• Equal Sign (=): Represents equality between two expressions.
• Not Equal Sign (≠): Indicates that two expressions are not equal.
• Approximately Equal Sign (≈): Used for values that are close but not exactly equal.
• Tilde (~): Denotes similarity or proportionality.
• Triple Bar (≡): Indicates identity or congruence in modular arithmetic.

Inequalities

• Less Than (<): Indicates that one quantity is smaller than another.
• Greater Than (>): Indicates that one quantity is larger than another.
• Less Than or Equal To (≤): Indicates that one value is smaller or equal to another.
• Greater Than or Equal To (≥): Same as above for greater values.
• Much Less Than (≪): Denotes a value significantly smaller.
• Much Greater Than (≫): Denotes a value significantly larger.

Set Theory Symbols

• Empty Set (∅): Denotes a set with no elements.
• Cardinality (#): Indicates the number of elements in a set.
• Membership (∈): Indicates that an element is a member of a set.
• Not In (∉): Indicates that an element is not a member.
• Subset (⊆): Represents that one set is a subset of another.
• Proper Subset (⊂): Denotes that sets are not equal.
• Union (∪): Combines two sets, containing all unique elements.
• Intersection (∩): Contains elements common to both sets.
• Set Difference (): Contains elements of the first set not in the second.
• Symmetric Difference (△ or ⊖): Contains elements in either set but not both.

Logic Symbols

• Negation (¬): Indicates the opposite of a statement.
• And (∧): True if both operands are true.
• Or (∨): True if at least one operand is true.
• Exclusive Or (⊕): True if exactly one operand is true.
• Logical Constants (T/F): T denotes true, F denotes false.
• Universal Quantifier (∀): Asserts a statement is true for all elements.
• Existential Quantifier (∃): Asserts at least one element satisfies a statement.
• Uniqueness Quantifier (∃!): Asserts exactly one element satisfies a statement.
• Conditional Operator (→): Denotes implication (if A then B).
• Logical Equivalence (↔): Indicates two statements have the same logical value.

Number Sets

• N: Set of natural numbers.
• Z: Set of integers.
• Q: Set of rational numbers.
• R: Set of real numbers.
• C: Set of complex numbers.
• H: Set of quaternions.
• O: Set of octonions.
• U: Universal set containing all possible values.

Calculus Notation

• Apostrophe (f'): Denotes the derivative of a function.
• Dots (ḟ): Represents the derivative of a variable with respect to time.
• Limit (lim): Denotes behavior of a function as input approaches a value.
• Integral (∫): Denotes anti-derivative, area under a curve.
• Partial Derivative (∂): Used for functions of several variables.

Additional Mathematical Symbols

• Infinity (∞): Represents unlimitedness, greater than any finite quantity.
• Cardinality of Infinite Sets (ℵ): For example, ℵ₀ for natural numbers.
• Factorial (!): Multiplies a number by all positive integers smaller than it.
• Binomial Coefficient (n choose k): Number of ways to choose k elements from n.
• Absolute Value (|x|): Distance of x from zero on the number line.
• 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: Returns the nearest integer to a given value.
• Visibility (|): Indicates divisibility; crossed line indicates non-divisibility.
• Parallelism (||): Two lines denote parallelism; crossed lines indicate non-parallelism.
• Perpendicularity (⊥): Indicates perpendicular lines or co-primality.
• Line Segment (A-B): Represents a line segment between points A and B.
• Array (→): Arrow from point A to B.
• Infinite Line (↔): Arrow pointing in both directions representing an infinite line.