# Relations

In mathematics, there are some relations such as “is less than”, “is perpendicular to”,

“is a power of ”.

A relation is any bonding between elements of one set, called the domain and another set,called the range.

A relation is a set of ordered pairs, so a binary relation consists of 3 components, domain, codomain and subset.

The first element in an ordered pair is from the domain and the second element from the range. The domain contains the independent variable and the range contains the
dependent variable. So we can say that the value of the range depends on
the domain.

__Example:__A= {Tina, Mona, Juli, Sera}

B={Mike, Jack, Tom}

Suppose Mike has two sisters Tina and Juli, Jack has one sister Mona, and Tom has one sister Sera.

If we define a relation R "is a sister of" between the elements of A and B then clearly,

If we define a relation R "is a sister of" between the elements of A and B then clearly,

Tina R Mike, Juli R Mike, Mona R Jack and Sera R Tom.

After removing R between two names these can be written in the form of ordered pairs as :

(Tina, Mike), (Juli, Mike), (Mona, Jack), (Sera, Tom).

The above information can be written in the form of a set R of ordered pairs as

R= {(Tom, Tina), (Jack, Tina), (Mike, Mona), (Jackson, Juli)}

Clearly in mathematics, if R $\subseteq $ A x B i.e. R = $\left \{ \left ( a,b \right ): a\in A, b\in B and aRb \right \}$ then R is said to be a relation.

If (a, b) ∈ R, we say that a is related to b under the relation R and we write as

a R b. And in relation, each value of the first set is paired with one or more values of the second set.

If R be a relation from A to B then,

- R = $\phi$, R is void relation.
- R=A × B, R is an universal relation.
- If R is a relation defined from A to A, it is called a relation defined on A
- R = { (a,a) $\forall$ a in A} , is called the identity relation.

The following is an algebraic relation that we will call b.

b:{(1,2) (3,4) (5,6) (7,8)}

The domain is: 1, 3, 5, 7 ( x values of the ordered pair)

The range is: 2, 4, 6, 8 ( y values of the ordered pair)

Here some examples of relations from A to B are:

- {(a, b) $\in$ A × B: a is uncle of b}.
- {(a, b) $\in$ A × B: a is father of b}.
- {(a, b) $\in$ A × B: age of a is less than age of b}.

So we can define mathematically, a relation R from A to B as an arbitrary subset of A × B.