# 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 arrow diagrams :**

**Tabular form representation**

### 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 2^{nd} 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 2^{p*q}.

n(A*B)=n(A)*n(B)=p*q so total number of subset of set A and B is 2^{p*q}. each subset of A*B is relation from A to B so total number of relation from A to B is 2^{p*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 2^{4} = 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)}