Combinatorics/Probabilities based on math for college(Математика, Богомолов, Самойленко, среднее профессиональное образование)

 

Combinatorics

Set

Collection of elements(in alg for example numbers)

Can be empty, singleton, finite and infinite sets

*inifinite set - for example, type of numbers(real, positive, etc…)

Sequence

Collection of numbers with a specific order(for example progressions)

*infinite set also a sequence(for example arithmetic progression)

Factorial

Product of all positive numbers

n! = n * (n-1) * (n-2) * (n-3) * … | while (n-x)>0

*for example 5! = 5 * 4 * 3 * 2 * 1 = 120

Elements of combinatorics

*possible actions with sets

Variations

From the set with n-length, it is possible to create new sets with rules - they have at least one different element, or order. Formula for calc all variations:

A(n m) = n*(n-1)*(n-2)*...(n-(m-1)) = n! / (n - m)!

Permutations

Special cases of variations when m=n, creates sets with the same length but with different order:

P(n) = A(n m) = n!

Combinations

Action of creating sets with at least one different element, the order do not have sense:

C(n m) = A(n m) / Pm = 1 / (n-m)!

Probabilities

Elements of probabilities

Event

Result of some action. Results can be predicted.

Random event

An event can be successful(true,1)/failure(false,0)/etc…

Certain event

Event will be a successful

Impossible event
Event will be a failure

Incompatible events

For multiple events: if first event is true then second is false, otherwise also

Independent events

For multiple: if first event true then second can be true/false 

Opposite events

For multiple: they describe one event and incompatible

Theorem of the incompatible probabilities sum

Probability of one successful from incompatible events 

P(A+B)=P(A)+P(B)

P(A1+A2+...Ak) = P(A1)+P(A2)+...+P(Ak)

Theorem of the independent probabilities sum

P(A+B)=P(A)+P(B)-P(AB)

*P(AB) probability when they both successful

P(A+B+C)=P(A)+P(B)+P(C)-P(AB)-P(AC)-P(BC)+P(ABC)

Theorem of the opposite probabilities sum

P(A)+P(B)=1

Theorem of the independent probabilities product

Probability when two successful from both

P(AB)=P(A)*P(B)

P(A1A2…An)=P(A1)*P(A2)*...*P(An)

Theorem of the incompatible probabilities product

Conditional probability

Probability of second(A) after first(B) is successful P(A/B)

P(AB)=P(A)*P(B/A)=P(B)*P(A/B)

integrals - based on high math handbook for students (Справочник по высшей математике для студентов вузов / Н. Г. Тактаров)

 

Antiderivative 

F(x) otherwise function to derivative function

If y’(x) = f(x) => y’(x) is F(x) 

Indefinite integral

∫ f(x)△x = F(x)+C

*C - random constant

*△x increment

Properties

∫ k*f(x)△x = k*∫ f(x)△x

∫ (f(x) +/- g(x))△x  = ∫ f(x)△x +/- ∫ g(x)△x  

Integral sum

∑[i=1,n] f(s) △x

*△x = x(i) - x(i-1)

*∑ - sum between

Definite integral

S = lim( △x->0 ) ∑[i=1,n] f(s) △x = ∫ [a,b] f(x)△x

*as we see geometrical understanding of definete integral is area under the plot

Properties

∫ [a,a] f(x)△x = 0

∫ [a,a] f(x)△x = - ∫ [a,a] f(x)△x

∫ [a,c] f(x)△x = ∫ [a,b] f(x)△x + ∫ [b,c] f(x)△x

∫ [a,b] (f(x) +/- g(x))△x = ∫ [a,b] f(x)△x +/- ∫ [a,b] g(x)△x

∫ [a,c] k*f(x)△x = k*∫ [a,c] f(x)△x

If f(x)<>=g(x) then ∫ [a,b] f(x)△x <>= ∫ [a,b] g(x)△x  

stereometry - based on stereometry chapter of handbook for BSATU abiturient (Математика абитуриенту: справочное пособие Минск : БГАТУ, 2008 сост Морозова)

crossection 

2D plane(shape) with intersection of 3D shape(figure) - crosssection of solid

Horizontal crossection

Vertical crossection

*of sphere

**also exist different/other types of crossections(with random angles, complex planes)

polyhedrons

polygon - shape built by line segments

Prism

Based on an n-angle polygon, with two connected parallel bases

*lines of connection are shaping parallelograms

Formulas

S(all) = 2*S(base)+ S(side)

S(side) - n*shape

V = S(of the base or horizontal crossection)*l

*l - side edge

*S - surface area

*V - volume 

Straight & regular prism

Diagonal in straight: d = a^2*b^2*l^2, l = h

*straigh with straigh angles to base

Cube

*all edges are equal

Pyramid

Straight and regular pyramid

*in straight - top vertex above the centroid of the base

Formulas

S(all) = S(sides)+S(base)

S(side) - S of triangle

V = ⅓*S(base)*h

*h - height

**for proof V:

Build cube with diagonals - will see 6 equal pyramids

Volume of cube = a*b*c; S(base) = a*b; c = H

In one pyramid h = H/2; H = 2*h; = > V = S*H/6; V = S*h*2/6 = S*h/3

Truncated pyramid

V = ⅓(S1+S2+sqrt(S1*S2))*h

*proof through coefficient of the big part and small part(they are common based) 

x(volume) = V - v(1)

V = ⅓*S*H

v = ⅓*s*h 

S/H = s/h

Sh=Hs

h = Hs/S - something like this, and to (1)  with H-h

curvilinear based

Cylinder

Prism wit a circle base

S(all)=2*S(base)+S(side)

S(side) = 2*pi*r*l

S(base)=pi*r^2

*horizontal crossection - circle

*vertical crossection - parallelogram

**ca be oblique(regular)

Cone

S(all)=S(side)+S(base)

S(side) = pi*l*r

S(base)=S(circle)

V=⅓*pi*r^2*h

*when roll-up the side it is not a circle but a sector and possible to find area through formules for the sector

Truncated cone

S(side) = (R + r)sqrt((R-r)^2+h^2)*pi

V=⅓*pi*h(R^2+r^2+R*r)

Sphere

3D version of circle - set of points with same distance from the center

S = 4*pi*R^2

V = 4/3*pi*r^3