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Relations

Relations between sets is a type of bond between elements of two or more sets. Relation can be defined as relation between element of sets. if set A is collection of distinct student of delhi public school and set B is collection of distinct teacher of delhi public school like A={Ram, Mohan, Sohan} set of student and set B={Abhay, Rahul} set of teacher then Relation R is a subset of A*B={(Ram, Abhay), (Mohan, Abhay), (Sohan, Abhay),(Ram, Rahul), (Mohan, Rahul), (Sohan, Rahul)}

Here (Ram, Abhay), (Mohan, Abhay), (Sohan, Abhay),(Ram, Rahul), (Mohan, Rahul), (Sohan, Rahul) is showing student and teacher relations. any subset of A*B is relation from A to B.

Examples: if set A={1, 2, 3} and set B={5, 6} A*B={(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6)} = R (relation from A to B)
R⊆A*B so R is relation from A to B. Here R represent the relation (is less then). (1, 5)∈R means 1 is less then 5. every subset of A*B denotes same relation.

Representation of a Relations

Relation from set A to set B can be represented through five different forms.
Roster form Representation of a Relation :Roster form of relation is set of all ordered pair belonging to relation R. roster form written as R={(a, b), (c, d)}

Set-builder form Representation of a Relation in set-builder form Representation relation is describe through some rule or property like one component is greater then another component (greater then relation between component ).
Example: R={(a, b): a∈A, b∈B, a is greater then b}
R={(x, y): x∈N, y∈N and x+y=8 }
R={(x, y): x∈N, y∈N and 2x+y=12 }

Lattice Representation of a Relation : Representation of a relation by lattice


Representation of a relation by arrow diagrams : Representation of a relation by arrow diagrams


Tabular form representation Representation of a relation by tabular form

Domain and Range of a relation

Domain of a Relation :If any relation from set A to set B is R then domain of relation R is defined as collection of all distinct element of relation where a ∈ A such that (a, b)∈ R for some b∈B. in other word domain of any relation is set of first component of all the ordered pairs belonging to R. domain of any relation R is written as Dom (R).

Example: if we have two set A={1, 2, 3} and B={k, m} and relation from A to B is R={(1, k), (2, k), (2, m), (3, k)} then domain of Relation R = set of first distinct components of all the ordered pairs belonging to R = {1, 2, 3}

Range of a Relation : If any relation from set A to set B is R then range of relation R is defined as collection of all distinct element of relation set where b ∈ B. in other word range of any relation is set of second component of all the ordered pairs belonging to R.

Example: if we have two set A={1, 2, 3} and B={k, m} and relation from A to B is R={(1, k), (2, k), (2, m), (3, k)} then range of Relation R = set of 2nd distinct components of all the ordered pairs belonging to R = {k, m}

Co-domain of a Relation : if R is any relation from A to B then B is known as co-domain of relation R.

Total number of relations

Total number of relations between two sets is defined as 2(number of element in first set * number of element in second set). if we have two non-empty finite sets A and B and number of element in first set is p and number of element in second set is q then total number of relation will be 2p*q.
n(A*B)=n(A)*n(B)=p*q so total number of subset of set A and B is 2p*q. each subset of A*B is relation from A to B so total number of relation from A to B is 2p*q

Example: if set A={4, 6} and set B={7, 8} and n(A*B)=n(A)*n(B)=2*2=4 then Total number of subset will be 24 = 16

inverse Relation

if any relation from set A to set B (R ⊆ A*B) is subset of A*B then inverse relation from set A to set B is denoted by R-1 is relation from set B to set A.

Example : if set A={2, 3} and set B={4, 5} relation from set A to set B is R={(2, 4), (3, 5)} then inverse relation will be written as R-1={(4, 2), (5, 3)}

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