Abbreviations
p.a. |
per annum |
NPV |
net present value |
resp. |
respectively |
rad |
radian |
llitre
mmetre
cm |
centimetre |
km |
kilometre |
s |
second |
EUR |
euro |
LPP |
linear programming problem |
s.t. |
subject to |
bv |
basic variables of a system of linear equations |
nbv |
non-basic variables of a system of linear equations |
Notations
A
A B
A B
A = B
A B
A(x)
x
A(x)
x
a A b / A
|A|
P(A) A B A B A ∩ B A \ B A × B
n
Ai
i=1
An
n!
n
k
n = 1, 2, . . . , k
N
N0
Z
negation of proposition A conjunction of propositions A and B disjunction of propositions A and B implication (if A then B) equivalence of propositions A and B universal proposition
existential proposition
a is an element of set A
b is not an element of set A empty set
cardinality of a set A (if A is a finite set, then |A| is equal to the number of elements in set A), the same notation is used for the determinant of a square matrix A
power set of set A
set A is a subset of set B union of sets A and B intersection of sets A and B difference of sets A and B
Cartesian product of sets A and B
Cartesian product of sets A1, A2, . . . , An
n
Cartesian product Ai , where A1 = A2 = . . . = An = A
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n factorial: n! = 1 · 2 · . . . · (n − 1) · n |
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binomial coefficient: |
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k = k |
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k) |
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! · |
− |
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equalities n = 1, n = 2, . . . , n = k
set of all natural numbers: N = {1, 2, 3, . . .} union of set N with number zero: N0 = N {0}
union of set N0 with the set of all negative integers
xvi List of notations |
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Q |
set of all rational numbers, i.e. set of all fractions p/q with p Z |
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and q N |
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R |
set of all real numbers |
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R+ |
set of all non-negative real numbers |
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(a, b) |
open interval between a and b |
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[a, b] |
closed interval between a and b |
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± |
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denotes two cases of a mathematical term: the first one with sign + |
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and the second one with sign − |
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denotes two cases of a mathematical term: the first one with sign − |
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|a| |
and the second one with sign + |
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absolute value of number a R |
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sign of approximate equality, e.g. √ |
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≈ 1.41 |
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≈ |
2 |
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= |
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sign ‘not equal’ |
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π |
irrational number equal to the circle length divided by the diameter |
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length: π ≈ 3.14159... |
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e |
Euler’s number: e ≈ 2.71828... |
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∞ |
infinity |
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√ |
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square root of a |
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e: y |
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exp(x) |
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ex |
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exp |
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= |
= |
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notation used for the exponential function with base y |
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log |
notation used for the logarithm: if y = loga x, then a |
= x |
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lg |
notation used for the logarithm with base 10: lg x = log10 x |
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ln |
notation used for the logarithm with base e: ln a = loge x |
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n |
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summation sign: |
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ai = a1 + a2 + · · · + an |
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i=1 |
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n |
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product sign: |
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2 |
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ai = a1 · a2 · |
. . . · an |
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i=1 |
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= −1 |
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i |
imaginary unit: i |
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C |
set of all complex numbers: z = a + bi, where a and b are real numbers |
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|z| |
modulus of number z C |
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{an} |
sequence: {an} = a1, a2, a3, . . . , an, . . . |
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{sn} |
series, i.e. the sequence of partial sums of a sequence {an} |
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lim |
limit sign |
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aRb |
a is related to b by the binary relation R |
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aRb |
a is not related to b by the binary relation R |
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R−1 |
inverse relation of R |
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S ◦ R |
composite relation of R and S |
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f : A → B |
mapping or function f A × B: f is a binary relation |
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b = f (a) |
which assigns to a A exactly one b B |
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b is the image of a assigned by mapping f |
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f −1 |
inverse mapping or function of f |
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g ◦ f |
composite mapping or function of f and g |
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Df |
domain of a function f of n ≥ 1 real variables |
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Rf |
range of a function f of n ≥ 1 real variables |
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y = f (x) |
y R is the function value of x R, i.e. the value of |
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function f at point x
deg P
x → x0
x → x0 + 0 x → x0 − 0
f(x), y (x)
f(x), y (x)
dy, df
≡
ρf (x0) εf (x0)
Rn Rn+ a
aT
|a|
|a − b| a b
dim V
Am,n
AT An
det A, (or |A|)
A−1 adj (A) r(A)
x1, x2, . . . , xn ≥ 0
Ri {≤, =, ≥}
z→ min!
z→ max!
fx (x0, y0)
fxi (x0)
grad f (x0)
ρf ,xi (x0) εf ,xi (x0)
Hf (x0)
List of notations xvii
degree of polynomial P x tends to x0
x tends to x0 from the right side x tends to x0 from the left side derivative of function f
derivative of function f with y = f (x) at point x second derivative of function f with y = f (x) at point x differential of function f with y = f (x)
sign of identical equality, e.g. f (x) ≡ 0 means that equality f (x) = 0 holds for any value x
proportional rate of change of function f at point x0 elasticity of function f at point x0
integral sign
n-dimensional Euclidean space, i.e. set of all real n-tuples set of all non-negative real n-tuples
vector: ordered n-tuple of real numbers a1, a2, . . . , an corresponding to a matrix with one column transposed vector of vector a
Euclidean length or norm of vector a
Euclidean distance between vectors a Rn and b Rn means that vectors a and b are orthogonal
dimension of the vector space V matrix of order (dimension) m × n transpose of matrix A
nth power of a square matrix A determinant of a matrix A inverse matrix of matrix A adjoint of matrix A
rank of matrix A
denotes the inequalities x1 ≥ 0, x2 ≥ 0, . . . , xn ≥ 0 means that one of these signs hold in the ith constraint of a system of linear inequalities
indicates that the value of function z should become minimal for the desired solution
indicates that the value of function z should become maximal for the desired solution
partial derivative of function f with z = f (x, y) with respect to x at point (x0, y0)
partial derivative of function f with z = f (x1, x2, . . . , xn) with respect to xi at point x0 = (x10, x20, . . . , xn0)
gradient of function f at point x0
partial rate of change of function f with respect to xi at point x0 partial elasticity of function f with respect to xi at point x0 Hessian matrix of function f at point x0
Q.E.D. (quod erat demonstrandum
– ‘that which was to be demonstrated’)