Lecture Notes on Mathematical Symbols and Notations

Basic Arithmetic Symbols

• Plus (+): Symbol for addition.
• Minus (-): Represents subtraction, also denotes negative numbers.
• Multiplication (×): Indicates multiplication; can be represented as a dot (·).
• Division (÷): Signifies division; may also be represented as a slash (/).

Sign Variations

• Plus-minus (±): Denotes either plus or minus; can indicate a range of values.
• Minus-plus (∓): Indicates the opposite sign of plus-minus.
• Root Symbol (√): Denotes square root; with a superscript, denotes N root of a number.

Equality Symbols

• Equal (=): Denotes equality between two expressions.
• Not equal (≠): Indicates that two expressions are not equal.
• Approximately equal (≈): Used when two values are not exactly equal but are close; can also use tilde (∼).
• Triple bar (≡): Denotes identity or congruence in modular arithmetic.

Inequality Symbols

• Less than (<): Indicates that one quantity is smaller than another.
• Greater than (>): Indicates that one quantity is larger than another.
• Less than or equal (≤): Indicates one value is smaller or equal to another.
• Greater than or equal (≥): Same as above, but for greater than.
• Much less than (≪) and Much greater than (≫): Denote extreme inequalities.

Set Theory Symbols

• Empty set (∅): Represents a set containing no elements.
• Membership (∈): Indicates that an element is a member of a set.
• Not in (∉): Indicates that an element is not a member of a set.
• Subset (⊆): Indicates one set is a subset of another; with a line (⊂) denotes proper subset.
• Union (∪): Combines two sets, resulting in a set of unique elements.
• Intersection (∩): Contains elements common in both sets.
• Set difference (): Contains all elements of the first set not in the second.
• Symmetric difference (⊕): Contains elements belonging to exactly one of the two sets.

Logic Symbols

• Negation (¬): Indicates the opposite of a statement.
• And (∧): Returns true if both operands are true.
• Or (∨): Returns true if at least one operand is true.
• Exclusive Or (⊕): Returns true if exactly one operand is true.
• T (True) and F (False): Constants for logical truth values.

Quantifiers

• Universal quantifier (∀): Asserts a statement is true for all elements in a domain.
• Existential quantifier (∃): Asserts there exists at least one element for which a statement holds true.
• Uniqueness quantifier (∃!): Asserts exactly one element exists for which a statement is true.

Functions and Derivatives

• Conditional operator (→): Denotes implication between two statements.
• Logical equivalence (↔): Indicates two statements have the same logical value.
• Derivative (f'): Denoted by an apostrophe, indicates the derivative of a function.
• Second derivative (f''): Denoted by adding another apostrophe.
• Dot notation: Used for derivatives with respect to time.
• Partial derivatives (∂): Denotes derivatives of functions with several variables.

Integrals and Limits

• Integral (∫): Denotes an anti-derivative.
• Definite integral (∫_a^b): Represents the area under a curve.
• Limit (lim): Denotes behavior of a function as input approaches a certain value._

Complex Numbers

• Real part (Re): Denotes the real part of a complex number.
• Imaginary part (Im): Denotes the imaginary part of a complex number.
• Complex conjugate (z̅): Changes the sign of the imaginary part.

Summation and Products

• Sigma (Σ): Denotes summation of a series of terms.
• Capital Pi (Π): Denotes a product of terms.

Infinity and Cardinality

• Infinity (∞): Represents unlimitedness.
• Aleph (ℵ): Represents the cardinality of infinite sets.
• Factorial (n!): Multiplies a number by all positive integers smaller than it.

Other Mathematical Operations

• Absolute value (|x|): Represents distance from zero on the number line.
• Floor function (⌊x⌋): Returns the greatest integer less than or equal to x.
• Ceiling function (⌈x⌉): Returns the smallest integer greater than or equal to x.

Geometry Symbols

• Parallel (∥): Denotes parallel lines; crossed lines denote non-parallelism.
• Perpendicular (⊥): Indicates perpendicular lines.
• Segment (AB): Denotes a line segment between points A and B.
• Ray (→AB): Starts at point A and extends through point B.
• Line (↔AB): Represents an infinite line passing through points A and B.