Contents 1 Mathematical logic 2 Sets 3 Miscellaneous signs and symbols 4 Operations 5 Functions 6 Exponential and logarithmic functions 7 Circular and hyperbolic functions 8 Complex numbers 9 Matrices 10 Coordinate systems 11 Vectors and tensors 12 Special functions 13 See also 14 References and notes

Mathematical logic Sign Example Name Meaning and verbal equivalent Remarks ∧ p ∧ q conjunction sign p and q ∨ p ∨ q disjunction sign p or q (or both) ¬ ¬ p negation sign negation of p; not p; non p ⇒ p ⇒ q implication sign if p then q; p implies q Can also be written as q ⇐ p. Sometimes → is used. ∀ ∀x∈A p(x) (∀x∈A) p(x) universal quantifier for every x belonging to A, the proposition p(x) is true The "∈A" can be dropped where A is clear from context. ∃ ∃x∈A p(x) (∃x∈A) p(x) existential quantifier there exists an x belonging to A for which the proposition p(x) is true The "∈A" can be dropped where A is clear from context. ∃! is used where exactly one x exists for which p(x) is true.

Sets Sign Example Meaning and verbal equivalent Remarks ∈ x ∈ A x belongs to A; x is an element of the set A ∉ x ∉ A x does not belong to A; x is not an element of the set A The negation stroke can also be vertical. ∋ A ∋ x the set A contains x (as an element) same meaning as x ∈ A ∌ A ∌ x the set A does not contain x (as an element) same meaning as x ∉ A { } {x1, x2, ..., xn} set with elements x1, x2, ..., xn also {xi ∣ i ∈ I}, where I denotes a set of indices { ∣ } {x ∈ A ∣ p(x)} set of those elements of A for which the proposition p(x) is true Example: {x ∈ ℝ ∣ x > 5} The ∈A can be dropped where this set is clear from the context. card card(A) number of elements in A; cardinal of A ∖ A ∖ B difference between A and B; A minus B The set of elements which belong to A but not to B. A ∖ B = { x ∣ x ∈ A ∧ x ∉ B } A − B should not be used. ∅ the empty set ℕ the set of natural numbers; the set of positive integers and zero ℕ = {0, 1, 2, 3, ...} Exclusion of zero is denoted by an asterisk: ℕ* = {1, 2, 3, ...} ℕk = {0, 1, 2, 3, ..., k − 1} ℤ the set of integers ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...} ℤ* = ℤ ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...} ℚ the set of rational numbers ℚ* = ℚ ∖ {0} ℝ the set of real numbers ℝ* = ℝ ∖ {0} ℂ the set of complex numbers ℂ* = ℂ ∖ {0} [,] [a,b] closed interval in ℝ from a (included) to b (included) [a,b] = {x ∈ ℝ ∣ a ≤ x ≤ b} ],] (,] ]a,b] (a,b] left half-open interval in ℝ from a (excluded) to b (included) ]a,b] = {x ∈ ℝ ∣ a < x ≤ b} [,[ [,) [a,b[ [a,b) right half-open interval in ℝ from a (included) to b (excluded) [a,b[ = {x ∈ ℝ ∣ a ≤ x < b} ],[ (,) ]a,b[ (a,b) open interval in ℝ from a (excluded) to b (excluded) ]a,b[ = {x ∈ ℝ ∣ a < x < b} ⊆ B ⊆ A B is included in A; B is a subset of A Every element of B belongs to A. ⊂ is also used. ⊂ B ⊂ A B is properly included in A; B is a proper subset of A Every element of B belongs to A, but B is not equal to A. If ⊂ is used for "included", then ⊊ should be used for "properly included". ⊈ C ⊈ A C is not included in A; C is not a subset of A ⊄ is also used. ⊇ A ⊇ B A includes B (as subset) A contains every element of B. ⊃ is also used. B ⊆ A means the same as A ⊇ B. ⊃ A ⊃ B. A includes B properly. A contains every element of B, but A is not equal to B. If ⊃ is used for "includes", then ⊋ should be used for "includes properly". ⊉ A ⊉ C A does not include C (as subset) ⊅ is also used. A ⊉ C means the same as C ⊈ A. ∪ A ∪ B union of A and B The set of elements which belong to A or to B or to both A and B. A ∪ B = { x ∣ x ∈ A ∨ x ∈ B } ⋃ ⋃ i = 1 n A i {\displaystyle \bigcup _{i=1}^{n}A_{i}} union of a collection of sets ⋃ i = 1 n A i = A 1 ∪ A 2 ∪ … ∪ A n {\displaystyle \bigcup _{i=1}^{n}A_{i}=A_{1}\cup A_{2}\cup \ldots \cup A_{n}} , the set of elements belonging to at least one of the sets A1, …, An. ⋃ i = 1 n {\displaystyle \bigcup {}_{i=1}^{n}} and ⋃ i ∈ I {\displaystyle \bigcup _{i\in I}} , ⋃ i ∈ I {\displaystyle \bigcup {}_{i\in I}} are also used, where I denotes a set of indices. ∩ A ∩ B intersection of A and B The set of elements which belong to both A and B. A ∩ B = { x ∣ x ∈ A ∧ x ∈ B } ⋂ ⋂ i = 1 n A i {\displaystyle \bigcap _{i=1}^{n}A_{i}} intersection of a collection of sets ⋂ i = 1 n A i = A 1 ∩ A 2 ∩ … ∩ A n {\displaystyle \bigcap _{i=1}^{n}A_{i}=A_{1}\cap A_{2}\cap \ldots \cap A_{n}} , the set of elements belonging to all sets A1, …, An. ⋂ i = 1 n {\displaystyle \bigcap {}_{i=1}^{n}} and ⋂ i ∈ I {\displaystyle \bigcap _{i\in I}} , ⋂ i ∈ I {\displaystyle \bigcap {}_{i\in I}} are also used, where I denotes a set of indices. ∁ ∁AB complement of subset B of A The set of those elements of A which do not belong to the subset B. The symbol A is often omitted if the set A is clear from context. Also ∁AB = A ∖ B. (,) (a, b) ordered pair a, b; couple a, b (a, b) = (c, d) if and only if a = c and b = d. ⟨a, b⟩ is also used. (,…,) (a1, a2, …, an) ordered n-tuple ⟨a1, a2, …, an⟩ is also used. × A × B cartesian product of A and B The set of ordered pairs (a, b) such that a ∈ A and b ∈ B. A × B = { (a, b) ∣ a ∈ A ∧ b ∈ B } A × A × ⋯ × A is denoted by An, where n is the number of factors in the product. Δ ΔA set of pairs (a, a) ∈ A × A where a ∈ A; diagonal of the set A × A ΔA = { (a, a) ∣ a ∈ A } idA is also used.

Miscellaneous signs and symbols Sign Example Meaning and verbal equivalent Remarks ≝   = d e f {\displaystyle \ {\stackrel {\mathrm {def} }{=}}} a ≝ b a is by definition equal to b [2] := is also used = a = b a equals b ≡ may be used to emphasize that a particular equality is an identity. ≠ a ≠ b a is not equal to b a ≢ b {\displaystyle a\not \equiv b} may be used to emphasize that a is not identically equal to b. ≙ a ≙ b a corresponds to b On a 1:106 map: 1 cm ≙ 10 km. ≈ a ≈ b a is approximately equal to b The symbol ≃ is reserved for "is asymptotically equal to". ∼ ∝ a ∼ b a ∝ b a is proportional to b < a < b a is less than b > a > b a is greater than b ≤ a ≤ b a is less than or equal to b The symbol ≦ is also used. ≥ a ≥ b a is greater than or equal to b The symbol ≧ is also used. ≪ a ≪ b a is much less than b ≫ a ≫ b a is much greater than b ∞ infinity () [] {} ⟨ ⟩ {\displaystyle \langle \rangle } (a+b)c [a+b]c {a+b}c ⟨ {\displaystyle \langle } a+b ⟩ {\displaystyle \rangle } c ac+bc, parentheses ac+bc, square brackets ac+bc, braces ac+bc, angle brackets In ordinary algebra, the sequence of (), [], {}, ⟨ ⟩ {\displaystyle \langle \rangle } in order of nesting is not standardized. Special uses are made of (), [], {}, ⟨ ⟩ {\displaystyle \langle \rangle } in particular fields.[3] ∥ AB ∥ CD the line AB is parallel to the line CD ⊥ {\displaystyle \perp } AB ⊥ {\displaystyle \perp } CD the line AB is perpendicular to the line CD[4]

Operations Sign Example Meaning and verbal equivalent Remarks + a + b a plus b − a − b a minus b ± a ± b a plus or minus b ∓ a ∓ b a minus or plus b −(a ± b) = −a ∓ b ... ... ... ... ⋮

Functions Example Meaning and verbal equivalent Remarks f : D → C {\displaystyle f:D\rightarrow C} function f has domain D and codomain C Used to explicitly define the domain and codomain of a function. f ( S ) {\displaystyle f\left(S\right)} { f ( x ) ∣ x ∈ S } {\displaystyle \left\{f\left(x\right)\mid x\in S\right\}} Set of all possible outputs in the codomain when given inputs from S, a subset of the domain of f. ⋮

Exponential and logarithmic functions Example Meaning and verbal equivalent Remarks e base of natural logarithms e = 2.718 28... ex exponential function to the base e of x logax logarithm to the base a of x lb x binary logarithm (to the base 2) of x lb x = log2x ln x natural logarithm (to the base e) of x ln x = logex lg x common logarithm (to the base 10) of x lg x = log10x ... ... ... ⋮

Circular and hyperbolic functions Example Meaning and verbal equivalent Remarks π ratio of the circumference of a circle to its diameter π = 3.141 59... ... ... ... ⋮

Complex numbers Example Meaning and verbal equivalent Remarks i   j imaginary unit; i² = −1 In electrotechnology, j is generally used. Re z real part of z z = x + iy, where x = Re z and y = Im z Im z imaginary part of z ∣z∣ absolute value of z; modulus of z mod z is also used arg z argument of z; phase of z z = reiφ, where r = ∣z∣ and φ = arg z, i.e. Re z = r cos φ and Im z = r sin φ z* (complex) conjugate of z sometimes a bar above z is used instead of z* sgn z signum z sgn z = z / ∣z∣ = exp(i arg z) for z ≠ 0, sgn 0 = 0

Matrices Example Meaning and verbal equivalent Remarks A matrix A ... ... ... ... ⋮

Coordinate systems Coordinates Position vector and its differential Name of coordinate system Remarks x, y, z [x y z] = [x y z]; [dx dy dz]; cartesian x1, x2, x3 for the coordinates and e1, e2, e3 for the base vectors are also used. This notation easily generalizes to n-mensional space. ex, ey, ez form an orthonormal right-handed system. For the base vectors, i, j, k are also used. ρ, φ, z [x, y, z] = [ρ cos(φ), ρ sin(φ), z] cylindrical eρ(φ), eφ(φ), ez form an orthonormal right-handed system. lf z= 0, then ρ and φ are the polar coordinates. r, θ, φ [x, y, z] = r [sin(θ)cos(φ), sin(θ)sin(φ), cos(θ)] spherical er(θ,φ), eθ(θ,φ),eφ(φ) form an orthonormal right-handed system.

Vectors and tensors Example Meaning and verbal equivalent Remarks a a → {\displaystyle {\vec {a}}} vector a Instead of italic boldface, vectors can also be indicated by an arrow above the letter symbol. Any vector a can be multiplied by a scalar k, i.e. ka. ... ... ... ⋮

Special functions Example Meaning and verbal equivalent Remarks Jl(x) cylindrical Bessel functions (of the first kind) ... ... ... ... ⋮