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Mathematics of Economics and Business.pdf
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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

 

 

 

i=1

n factorial: n! = 1 · 2 · . . . · (n 1) · n

binomial coefficient:

k = k

 

(n !

k)

 

n

 

n

 

 

 

! ·

 

!

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

 

 

 

 

 

 

 

 

 

Q

set of all rational numbers, i.e. set of all fractions p/q with p Z

 

 

 

 

 

and q N

 

 

 

 

 

 

 

 

 

R

set of all real numbers

 

 

 

 

 

R+

set of all non-negative real numbers

 

 

 

 

 

(a, b)

open interval between a and b

 

 

 

 

 

[a, b]

closed interval between a and b

 

 

 

 

 

±

 

 

denotes two cases of a mathematical term: the first one with sign +

 

 

and the second one with sign

 

 

 

 

 

denotes two cases of a mathematical term: the first one with sign

 

|a|

and the second one with sign +

 

 

 

 

 

absolute value of number a R

 

 

 

 

 

 

 

 

sign of approximate equality, e.g.

 

1.41

 

 

 

 

 

2

 

 

 

 

 

=

 

 

sign ‘not equal’

 

 

 

 

 

 

 

 

 

π

irrational number equal to the circle length divided by the diameter

 

 

 

 

length: π 3.14159...

 

 

 

 

 

e

Euler’s number: e 2.71828...

 

 

 

 

 

infinity

 

 

 

 

 

 

 

 

 

a

 

square root of a

 

 

 

e: y

 

exp(x)

 

ex

exp

 

 

 

 

=

=

 

 

 

notation used for the exponential function with base y

 

 

 

log

notation used for the logarithm: if y = loga x, then a

= x

 

 

 

lg

notation used for the logarithm with base 10: lg x = log10 x

 

 

 

ln

notation used for the logarithm with base e: ln a = loge x

 

 

 

 

 

 

n

 

 

 

 

 

 

 

 

 

 

 

 

summation sign:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ai = a1 + a2 + · · · + an

 

 

 

 

 

 

 

i=1

 

 

 

 

 

 

 

 

 

 

n

 

 

 

 

 

 

 

 

 

 

 

 

product sign:

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

ai = a1 · a2 ·

. . . · an

 

 

 

 

 

 

 

 

i=1

 

= −1

 

 

 

 

 

i

imaginary unit: i

 

 

 

 

 

 

C

set of all complex numbers: z = a + bi, where a and b are real numbers

|z|

modulus of number z C

 

 

 

 

 

{an}

sequence: {an} = a1, a2, a3, . . . , an, . . .

 

 

 

 

 

{sn}

series, i.e. the sequence of partial sums of a sequence {an}

 

 

 

lim

limit sign

 

 

 

 

 

 

 

 

 

aRb

a is related to b by the binary relation R

 

 

 

 

 

aRb

a is not related to b by the binary relation R

 

 

 

 

 

R1

inverse relation of R

 

 

 

 

 

S R

composite relation of R and S

 

 

 

 

 

f : A B

mapping or function f A × B: f is a binary relation

 

 

 

 

 

b = f (a)

which assigns to a A exactly one b B

 

 

 

 

 

b is the image of a assigned by mapping f

 

 

 

 

 

f 1

inverse mapping or function of f

 

 

 

 

 

g f

composite mapping or function of f and g

 

 

 

 

 

Df

domain of a function f of n 1 real variables

 

 

 

 

 

Rf

range of a function f of n 1 real variables

 

 

 

 

 

y = f (x)

y R is the function value of x R, i.e. the value of

 

 

 

 

 

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|)

A1 adj (A) r(A)

x1, x2, . . . , xn 0

Ri {≤, =, ≥}

zmin!

zmax!

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’)

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