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..
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