Sunday 13 May 2018
Wednesday 20 September 2017
PROBABILITY ( प्रायिकता )
PROBABILITY- in 1993 A.N. Kolmogrov, a Russian mathematician tried succesfully to relate the theory of probability eith the set theory by axiomatic aproch.
Random variables - A real valued function defined on a sample -space is called a randon - variable .
SAMPLE SPACE- A sample space of a random experiment is the set of all possible outcomes of that experiment and is denoted by S.
for example- if a coin is tossed then there are two possibilities either we shall get a head or tail . we denoted here head (H) and tail(T).
EXHAUSTIVE EVENTS- All possible outcomes in a trial are called exhaustive events.
SAMPLE POINT:- Every element of the sample space is called a sample point. and it it is contains finite number of point then its is called finite sample point.for example in a coin H and T.
EVENTS:- Of all the possible outcomes in the sample space of an experiment some outcomes satisfy a specified description , it is called an event and its denoted by E.
CERTAINS AND IMPOSSIBLE EVENTS:- if S is a sample space then S and Φ are both subset of S and both are events.so S is called certains events and Φ is called impossible events.
EQUALLY LIKELY EVENTS:- Two wvents are considered equally likely if one of them cannot ve expected in preference to the other.
EXHAUSTIVE EVENT:- All possible outcomes in a trial are called exhaustive events.
for example :- if we trail a coin then the exhaustive events are H and T.
MUTUALLY EVENT OR INCOMPITIBLE EVENT:- Two or more than two evetns are called Mutually exclusive events if there are no element common to these events .
if E1 and E2 are two mutually exclusive events then E1 =E2 =Φ
EXAMPLES :- An experiment in which two coins are tossed together fine the sample space.
solve- S={(H,H) (H,T) (T,T) (T,H)}
IF coins tossed tree times then S ={ (HHH, TTT, THH, HHT, HTT, THT,}
SAMPLE OF COMPOUND EVENTS:- If E contains only one elements of the sample space S then E is called simple events i.e E= {ei}
compound events :- If E contains more than one elements of the sample space S then S is called compound event.
E={ei} where i = 1,2,3....n
FORMULLE FOR PROBABILITY OF AN EVENT:- let E be an event of S containing m element of S ie n(E) = m if P(E) is the probability of the event E happening then P(E) = n(E) /n(S)
COMPLEMENTARY EVENT:- if E be an event then not happining of the event E is called the complementary event of E and is denoted by E'.
and p(E) + p(E') = 1
Q:- find the probability of throwing on even number with a die.
solve :- let S be a sample space and the event of getting an event number be E then
S ={ 1,2,3,4,5,6} and E={2,4,6}
so n(S) = 6 and n(E) = 3 [ for even number]
therefore , P(E) = n(E) /n(S)
= 3/6
= 1/2
Q:- if two coin tossed , find the chance that there should be heads on both.
solve:- if two coins tossed then sample space is
S={(H,H) (H,T) (T,T) (T,H)}
both head have only one condition that is ( H,H)
therefore P(E) = n(E) /n(S)
= 1/4
Q:- Find the probability of throwing on even number with die.
solve:-let S be the sample space and the event of getting an even number be E then
S = {1,2,3,4,5,6} and E={2,4,6}
n(S) = 6 and n(E) = 3
therefore the probability of event of happening p(E)=n(E)/n(S)
= 3/6
=1/2 =ans
COMPOSITION OF EVENTS:-
THE EVENT REPRESENTED BY AUB OR A+B:-- IF the event E happens when A happens or B happens then E is denoted by and E is represented by AUB i.e. E=AUB
THE EVENT REPRESENTED BY A ∩ B OR AB:--If The event E happens when the events A and B both happens then the events E is represented by A ∩ B OR AB i.e. E=A ∩ B
COMPLEMENT OF EVENT A OR THE EVENT A':-- If the event E happpens when the event A does not happen then E is denoted by A'
Theoram :- If E1 and E2 are any two events then P(E1 U E2) = P(E1)+P(E2)-P(E1∩ E2).
PROOF:-
lets S be the sample space and n be the number of elements in the events in S.
let l be the number of elements in E1 and m the number of elements in the events E2
i.e. n(S)= n n(E2)= l n(E2)=m
if the events E1 and E2 are not mutually exclusive then the E1∩ E2 is not equal to phy.
let n(E1∩ E2) = r
clearly, n(E1U E2) = l+m-r
now the probability of E1 and E2 happening denoted by P(E1∩ E2) is given by
p(E1U E2) =
Random variables - A real valued function defined on a sample -space is called a randon - variable .
SAMPLE SPACE- A sample space of a random experiment is the set of all possible outcomes of that experiment and is denoted by S.
for example- if a coin is tossed then there are two possibilities either we shall get a head or tail . we denoted here head (H) and tail(T).
EXHAUSTIVE EVENTS- All possible outcomes in a trial are called exhaustive events.
SAMPLE POINT:- Every element of the sample space is called a sample point. and it it is contains finite number of point then its is called finite sample point.for example in a coin H and T.
EVENTS:- Of all the possible outcomes in the sample space of an experiment some outcomes satisfy a specified description , it is called an event and its denoted by E.
CERTAINS AND IMPOSSIBLE EVENTS:- if S is a sample space then S and Φ are both subset of S and both are events.so S is called certains events and Φ is called impossible events.
EQUALLY LIKELY EVENTS:- Two wvents are considered equally likely if one of them cannot ve expected in preference to the other.
EXHAUSTIVE EVENT:- All possible outcomes in a trial are called exhaustive events.
for example :- if we trail a coin then the exhaustive events are H and T.
MUTUALLY EVENT OR INCOMPITIBLE EVENT:- Two or more than two evetns are called Mutually exclusive events if there are no element common to these events .
if E1 and E2 are two mutually exclusive events then E1 =E2 =Φ
EXAMPLES :- An experiment in which two coins are tossed together fine the sample space.
solve- S={(H,H) (H,T) (T,T) (T,H)}
IF coins tossed tree times then S ={ (HHH, TTT, THH, HHT, HTT, THT,}
SAMPLE OF COMPOUND EVENTS:- If E contains only one elements of the sample space S then E is called simple events i.e E= {ei}
compound events :- If E contains more than one elements of the sample space S then S is called compound event.
E={ei} where i = 1,2,3....n
FORMULLE FOR PROBABILITY OF AN EVENT:- let E be an event of S containing m element of S ie n(E) = m if P(E) is the probability of the event E happening then P(E) = n(E) /n(S)
COMPLEMENTARY EVENT:- if E be an event then not happining of the event E is called the complementary event of E and is denoted by E'.
and p(E) + p(E') = 1
Q:- find the probability of throwing on even number with a die.
solve :- let S be a sample space and the event of getting an event number be E then
S ={ 1,2,3,4,5,6} and E={2,4,6}
so n(S) = 6 and n(E) = 3 [ for even number]
therefore , P(E) = n(E) /n(S)
= 3/6
= 1/2
Q:- if two coin tossed , find the chance that there should be heads on both.
solve:- if two coins tossed then sample space is
S={(H,H) (H,T) (T,T) (T,H)}
both head have only one condition that is ( H,H)
therefore P(E) = n(E) /n(S)
= 1/4
Q:- Find the probability of throwing on even number with die.
solve:-let S be the sample space and the event of getting an even number be E then
S = {1,2,3,4,5,6} and E={2,4,6}
n(S) = 6 and n(E) = 3
therefore the probability of event of happening p(E)=n(E)/n(S)
= 3/6
=1/2 =ans
COMPOSITION OF EVENTS:-
THE EVENT REPRESENTED BY AUB OR A+B:-- IF the event E happens when A happens or B happens then E is denoted by and E is represented by AUB i.e. E=AUB
PROOF:-
lets S be the sample space and n be the number of elements in the events in S.
let l be the number of elements in E1 and m the number of elements in the events E2
i.e. n(S)= n n(E2)= l n(E2)=m
if the events E1 and E2 are not mutually exclusive then the E1∩ E2 is not equal to phy.
let n(E1∩ E2) = r
clearly, n(E1U E2) = l+m-r
now the probability of E1 and E2 happening denoted by P(E1∩ E2) is given by
p(E1U E2) =
hence proved
Tuesday 22 August 2017
GRAPH
We study here some type of graph like multigraph ,multigraph,path,circuit,etc...
now we define here a graph. start as...
DIRECTED GRAPH:- A directed graph is a graph defined abstractly as an ordred pair (V,E) where V={v1,v2,.....} vertex and E={e1,e2,e3.........} i.e. egdes is a binary relation on V. The pair (vi,vj) is said to be incedent vertices.where vi is a initial vertax and vj is a terminal vertix.
SELF LOOP:- The defination of a graph an edge to be of the form (vi,vj) suvh an edge having the same vertex as both its end vertices is called a self loop
where K is a self loop
PARALLEL EDGES IN A GRAPH:- Let G=(V,E) is a graph then all edges having the same pair of end vertices are called parallel edges ..
where e1 and e2 are parallel edges.
UNDIRECTED GRAPH:- An undirected graph is defined abstractly as an ordered pair (V,E) where V is a non-empty set and E is a multisets of two elements from V.for example
this is an undirected graph because it have not any direction.
SIMPLE GRAPH:- A graph G=(V,E) that has neither self -loop nor parallel edges is called a simple graph. for example
it have no self loop and no parallel edges.
FINITE AND INFINITE GRAPH:- A graph with a finite number of vertices as well as a finite number of edges is called a finite graph otherwise it is infinite graph.
this is a finite graph because it have finite number of vertices and edges.
this is an infinite graph.
ORDER OF A GRAPH:- If G=(V,E) is a fiinite group then the number of vertices is is called the order of the graph G and its denoted by IVI (mod V).
INCEDENCE :- Let E be and edge joining two vertices vi to vj of a graph G=(V,E) then the edge e is said to be incident on each of its end vertices vi to vj.
ADJACENCY:- Two vertices in a graph are said to ve adjacent if there are exist an edge joining the vertices.
DEGREE OF A VERTEX:- The degree of a vertex v in a graph written as d(v) is equal to the number of edges which are incident on v with self -loop counted twice.
In fig A d(a)=2 , d(b)=2 , d(C)= 2
ISOLATED VERTEX:- a vertex in a graph G having no edge incedent on it is called an isolated vertex.
in this graph f is a isolated vertex.
PENDANT VERTEX:- A vertex v in a graph G is said to ve pendent vertex if its degree is one. i.e. d(v) = 1 . in fig B degree of e is 1.
NULL GRAPH:- A graph is said to be null if its each vertex is zero degree or other word its vertices are non empty but its edes are empty.
EVEN OR ODD VERTICES:- A vertex is said to be even or odd if its degree is an even of odd number.
THEORAM:- The sum of the degrees of all vertices in a graph G is equal to twice the number of edges in G.
solve:- Let G=(V,E) be a graph ,then the number of edges in Gis IEI,
since the edge in G is incident on two vertices so it contributes 2 to the sum of the degrees of the graph ,
so sum of degrees of all the vertices in G = ∑d(v)=2IEI
this theoram is called the HANDSHAKING LEMMA.
FOR EXAMPLE:-
now we define here a graph. start as...
DIRECTED GRAPH:- A directed graph is a graph defined abstractly as an ordred pair (V,E) where V={v1,v2,.....} vertex and E={e1,e2,e3.........} i.e. egdes is a binary relation on V. The pair (vi,vj) is said to be incedent vertices.where vi is a initial vertax and vj is a terminal vertix.
SELF LOOP:- The defination of a graph an edge to be of the form (vi,vj) suvh an edge having the same vertex as both its end vertices is called a self loop
where K is a self loop
PARALLEL EDGES IN A GRAPH:- Let G=(V,E) is a graph then all edges having the same pair of end vertices are called parallel edges ..
UNDIRECTED GRAPH:- An undirected graph is defined abstractly as an ordered pair (V,E) where V is a non-empty set and E is a multisets of two elements from V.for example
SIMPLE GRAPH:- A graph G=(V,E) that has neither self -loop nor parallel edges is called a simple graph. for example
 |
| FIG A |
FINITE AND INFINITE GRAPH:- A graph with a finite number of vertices as well as a finite number of edges is called a finite graph otherwise it is infinite graph.
this is a finite graph because it have finite number of vertices and edges.
ORDER OF A GRAPH:- If G=(V,E) is a fiinite group then the number of vertices is is called the order of the graph G and its denoted by IVI (mod V).
INCEDENCE :- Let E be and edge joining two vertices vi to vj of a graph G=(V,E) then the edge e is said to be incident on each of its end vertices vi to vj.
ADJACENCY:- Two vertices in a graph are said to ve adjacent if there are exist an edge joining the vertices.
DEGREE OF A VERTEX:- The degree of a vertex v in a graph written as d(v) is equal to the number of edges which are incident on v with self -loop counted twice.
In fig A d(a)=2 , d(b)=2 , d(C)= 2
ISOLATED VERTEX:- a vertex in a graph G having no edge incedent on it is called an isolated vertex.
 |
| fig B |
PENDANT VERTEX:- A vertex v in a graph G is said to ve pendent vertex if its degree is one. i.e. d(v) = 1 . in fig B degree of e is 1.
NULL GRAPH:- A graph is said to be null if its each vertex is zero degree or other word its vertices are non empty but its edes are empty.
THEORAM:- The sum of the degrees of all vertices in a graph G is equal to twice the number of edges in G.
solve:- Let G=(V,E) be a graph ,then the number of edges in Gis IEI,
since the edge in G is incident on two vertices so it contributes 2 to the sum of the degrees of the graph ,
so sum of degrees of all the vertices in G = ∑d(v)=2IEI
this theoram is called the HANDSHAKING LEMMA.
FOR EXAMPLE:-
Friday 18 August 2017
RELATION : BINARY RELATION AND POSET
RELATION:- In order to express a relation from set A to Set B , we always need a statement which connects the elements of A with the elements of B.
example ;-A ={1,2,3,4} B={2,4,6} and relation is less than than we express it like (1,2) (1,4) (1,6) (2,4) (2,6) (3,4) (4,6). that is 1 is lass than 2 , 1 is less than 4 ...etc
BINARY RELATION -
let A and B be two non-empty sets . A relation from A to B is a subset of A-B and is denoted by R .R is a relation from A to B such that R ⊆ A-B
R = {(a,b) : a∈A , b∈B and aRb}
example ;-A ={1,2,3,4} B={2,4,6} and relation is less than than we express it like (1,2) (1,4) (1,6) (2,4) (2,6) (3,4) (4,6). that is 1 is lass than 2 , 1 is less than 4 ...etc that is R ⊆ A-B we can write this like R ⊆ AxB
memory :- if Set A and B has m and m elements that we write it total number of elements is m.m or squre of m.
if Set A has m and set B has n elements than we write is m.n elements
INVERSE RELATION:-if R ⊆ AxB is a relation from A to B then inverse relation define as R-1= { (b,a):(a,b)∈A a∈A , b∈B and aRb}
COMPOSITE RELATION:- Let A , B,C be three non-empty sets and R be a relation from A to B and S be a relation from B to C st R ⊆ AxB and S ⊆ BxC then the composite relation of two relations is a relation from A to C define as SoR ={(a,b) : an element c∈B st (a,c)∈R , (c,b)∈S}
i.e. (a,c)∈R , (c,b)∈S=> (a,b)∈ SoR .
Rusult:- if R-1 and S-1 are inverse of the relations R and S respectively then
(RoS)-1 = S-1 o R-1
RELATION ON A SET :- Let R be a relation from P to Q . if P =Q then we say that R is a relation on a Set P. i.e R ⊆ PxQ.
TYPES OF BINARY RELATIONS:-
1. Reflexive:- Let A be a non- empty set and R be a binary relation in A i.e. R ⊆ AxA then the relation R is called relexive relation if every element of A is R related to itself.
i.e. for all (a,a)∈R
2. Symmetric :-
If R is a relation in the set A then R is called symmetric relation if a is related to a then a is related to b.
i.e. (a,b)∈R => (b,a)∈R.
3. Anty-symmetric Relation :-
If R is a relation in the set A , then B is called anty-symmetric if (a,b)∈R (b,a)∈R. => a=b
4. Transitive Relation :-
If R is a relation in the set A then R is called transitive if a is related to b and b related to c then a is related to c i.e.
(a,b)∈R (b,c)∈R =>(a,c)∈R.
Partial Order set (poset):- A relation R on a set A is said to be Partial order set if it is Reflexice , Antisymmtric and Transitive. And if it satisfies another axiom trichnotomy low then it is called total ordering relation. trichnotomy law is any two element of set is comparble.
compleltely defination of Partial order set is :- A set A with a partial ordering relation≤on A is called partial order set and its denoted by( A,≤)
for Example:- If N be the set of partial integers then prove that relation ≤ is a partial order set on N.
solve:-
1. Reflexivity:- for each a∈N . since a is less or equal to itself then we can say a≤a. i.e. aRa.
2. Anti-symmtry:- let a,b∈N such that a≤b , b≤a so no element cannot be less or equal each other. therefore a=b.
3.Transitivity:- let a,b,c∈N st a≤b , b≤c then a≤c.
and hence it is a partial order set. hence proved.
Maximal element:- let ( A,≤) be a partially order set .An element a in A is called maximal element if
Minimal element:- let ( A,≤) be a partially order set .An element p in A is called maximal element if x≤ p => m=x.
Least Upper Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) x≤a for all x∈S.
2) there are exist a Upper Bound t such that a≤t.
I.e. lub is less or equal all other upper bounds.
Greatest lower Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) a≤x for all x∈S.
2) there are exist a lower Bound w such that w≤a.
I.e. glb is greater or equal all other lower bounds.
LATTICE:- A partially ordered set ( L,≤) is said to be lattice if evety two elements in the set L has a unique least upper bound and a unique greatest lower bound or its inf or sup exist.
DUAL LATTICE:- let ( L,≤) be a partial order set for any two element a,b∈L the converse of the relation "≤ " denoted by "≥" defined as b≥a <=>a≤b . in other words ( L,≤) is a lattice then ( L,≥) also a lattice.
continue..
example ;-A ={1,2,3,4} B={2,4,6} and relation is less than than we express it like (1,2) (1,4) (1,6) (2,4) (2,6) (3,4) (4,6). that is 1 is lass than 2 , 1 is less than 4 ...etc
BINARY RELATION -
let A and B be two non-empty sets . A relation from A to B is a subset of A-B and is denoted by R .R is a relation from A to B such that R ⊆ A-B
R = {(a,b) : a∈A , b∈B and aRb}
example ;-A ={1,2,3,4} B={2,4,6} and relation is less than than we express it like (1,2) (1,4) (1,6) (2,4) (2,6) (3,4) (4,6). that is 1 is lass than 2 , 1 is less than 4 ...etc that is R ⊆ A-B we can write this like R ⊆ AxB
memory :- if Set A and B has m and m elements that we write it total number of elements is m.m or squre of m.
if Set A has m and set B has n elements than we write is m.n elements
INVERSE RELATION:-if R ⊆ AxB is a relation from A to B then inverse relation define as R-1= { (b,a):(a,b)∈A a∈A , b∈B and aRb}
COMPOSITE RELATION:- Let A , B,C be three non-empty sets and R be a relation from A to B and S be a relation from B to C st R ⊆ AxB and S ⊆ BxC then the composite relation of two relations is a relation from A to C define as SoR ={(a,b) : an element c∈B st (a,c)∈R , (c,b)∈S}
an element c∈B st (a,c)∈R , (c,b)∈S}i.e. (a,c)∈R , (c,b)∈S=> (a,b)∈ SoR .
Rusult:- if R-1 and S-1 are inverse of the relations R and S respectively then
(RoS)-1 = S-1 o R-1
RELATION ON A SET :- Let R be a relation from P to Q . if P =Q then we say that R is a relation on a Set P. i.e R ⊆ PxQ.
TYPES OF BINARY RELATIONS:-
1. Reflexive:- Let A be a non- empty set and R be a binary relation in A i.e. R ⊆ AxA then the relation R is called relexive relation if every element of A is R related to itself.
i.e. for all (a,a)∈R
2. Symmetric :-
If R is a relation in the set A then R is called symmetric relation if a is related to a then a is related to b.
i.e. (a,b)∈R => (b,a)∈R.
3. Anty-symmetric Relation :-
If R is a relation in the set A , then B is called anty-symmetric if (a,b)∈R (b,a)∈R. => a=b
4. Transitive Relation :-
If R is a relation in the set A then R is called transitive if a is related to b and b related to c then a is related to c i.e.
(a,b)∈R (b,c)∈R =>(a,c)∈R.
Partial Order set (poset):- A relation R on a set A is said to be Partial order set if it is Reflexice , Antisymmtric and Transitive. And if it satisfies another axiom trichnotomy low then it is called total ordering relation. trichnotomy law is any two element of set is comparble.
compleltely defination of Partial order set is :- A set A with a partial ordering relation≤on A is called partial order set and its denoted by( A,≤)
for Example:- If N be the set of partial integers then prove that relation ≤ is a partial order set on N.
solve:-
1. Reflexivity:- for each a∈N . since a is less or equal to itself then we can say a≤a. i.e. aRa.
2. Anti-symmtry:- let a,b∈N such that a≤b , b≤a so no element cannot be less or equal each other. therefore a=b.
3.Transitivity:- let a,b,c∈N st a≤b , b≤c then a≤c.
and hence it is a partial order set. hence proved.
Maximal element:- let ( A,≤) be a partially order set .An element a in A is called maximal element if
Minimal element:- let ( A,≤) be a partially order set .An element p in A is called maximal element if x≤ p => m=x.
Least Upper Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) x≤a for all x∈S.
2) there are exist a Upper Bound t such that a≤t.
I.e. lub is less or equal all other upper bounds.
Greatest lower Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) a≤x for all x∈S.
2) there are exist a lower Bound w such that w≤a.
I.e. glb is greater or equal all other lower bounds.
LATTICE:- A partially ordered set ( L,≤) is said to be lattice if evety two elements in the set L has a unique least upper bound and a unique greatest lower bound or its inf or sup exist.
DUAL LATTICE:- let ( L,≤) be a partial order set for any two element a,b∈L the converse of the relation "≤ " denoted by "≥" defined as b≥a <=>a≤b . in other words ( L,≤) is a lattice then ( L,≥) also a lattice.
continue..
Monday 14 August 2017
Vector space
Vector space:-
Let V be a set with operation + and let F be a field with the operations + and . (dot) . an algebric expression ((V,+),(F,+.), .) with the internal and external operations is called vector space if it is satisfies following axioms.
1. (V,+) be an abelian group.
2. (F,+,.) Be closer with respect to dot.
3. a(æ+ß) = aæ+aß for all a,b is in F and æ,ß in V.
In other words we can express it as below..
we can describe this
1. (V,+) be an abelian group i,e, it satisfies 5 charactiristic closure associate identity inverse and commutative /
1 α,β in V then α+β in V.
2 α,β,γ in V then (α+β)+γ= α+(β+γ)
3 α in V , then there are exist 0 st α+0 = α
4 if α in V then -α in V st α+(α) in V.
5 α,β in V then α+β =β+α
2. (F,+,.) Be closer with respect to dot. i,e,
a in F and α in V st aα in V.
* set of real no is a vector space .
Let V be a set with operation + and let F be a field with the operations + and . (dot) . an algebric expression ((V,+),(F,+.), .) with the internal and external operations is called vector space if it is satisfies following axioms.
1. (V,+) be an abelian group.
2. (F,+,.) Be closer with respect to dot.
3. a(æ+ß) = aæ+aß for all a,b is in F and æ,ß in V.
In other words we can express it as below..
we can describe this
1. (V,+) be an abelian group i,e, it satisfies 5 charactiristic closure associate identity inverse and commutative /
1 α,β in V then α+β in V.
2 α,β,γ in V then (α+β)+γ= α+(β+γ)
3 α in V , then there are exist 0 st α+0 = α
4 if α in V then -α in V st α+(α) in V.
5 α,β in V then α+β =β+α
2. (F,+,.) Be closer with respect to dot. i,e,
a in F and α in V st aα in V.
* set of real no is a vector space .
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