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.
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,
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,
  1. R = $\phi$, R is void relation.
  2. R=A × B, R is an universal relation.
  3. If R is a relation defined from A to A, it is called a relation defined on A
  4. 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:
  1. {(a, b) $\in$ A × B: a is uncle of b}.
  2. {(a, b) $\in$ A × B: a is father of b}.
  3. {(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.

This is the expansion of identity relation. In reflexive relation every element of the set is related to itself. So we say that,
R is reflexive if, (a, a) $\in$ R, for every a $\in$ A.

Consider a set A={a,b,c}. Then, one of the possible reflexive relations can be :
R={(a, a),(b, b),(c, c),(a, b),(a, c)}

But, it is not a reflexive relation as the condition is not fulfill by the subset.
So the condition of reflexive relation is
R is reflexive $\Leftrightarrow$ (a,a)$\in$ R $\forall $ a$\in$ A.

In some condition identity relation is a reflexive relation. Universal relation consists of all combinations of ordered pairs in the Cartesian product other ordered pairs. So, universal relation is also a reflexive relation.

The relations “is equal to”, “is less than or equal to”, “is greater than or equal to”, “divides”, “is subset of” are reflexive relations.

To show that, “less than or equal to” is a reflexive relation for natural number?
A relation, “R”, representing “less than or equal to” is defined as relation on natural number as :
(x,y) $\in$ R $\Leftrightarrow$ x$\leq$ y ,where y$\in$ N.

We construct data for “x” and “y” in accordance with the given relation for few initial natural numbers, say 1, 2 and 3, as under :
For x=1, y=1,2,3
For x=2, y=2,3
For x=3, y=3

Thus, the relation set is :
It is clear that, R consists of relation of all elements of the set, which are related to itself i.e. (1,1), (2,2) and (3,3). So the given relation is a reflexive relation.
In symmetric relation, ordered pairs has its mirror image.
A binary relation R over a set X is symmetric if,
(a,b) $\in$ R $\Rightarrow$ (b,a) $\in$ R $\forall$ a,b $\in$ X.

We know that, order of elements in relation is not important. In order to decide symmetry of a relation, we need to identify mirror pairs.

Alternatively, the condition of symmetric relation can be stated as :
aRb $\Rightarrow$ bRa $ \forall$ a,b$\in$A
In words, we say that if (a, b) be an instance of a relation, then (b, a) will also be the instance of a symmetric relation "R".
It is clear that identity relation and universal relation both are symmetric relation.

The relations "is married to", "is equal to", "is comparable to" are symmetric relation.
Let “R” be the relation on set A, then the condition of transitive relation as :
If (a, b)$\in$ R and (b, c)$\in$ R then (a, c)$\in$ R $\forall$ a,b,c$\in$ A.

We can say if,
aRb and bRc $\Rightarrow$ aRc for all a, b, c $\in$∈A.

In other words, we say that if (a, b) and (b, c) be related under R, such that (a,c) is also related under R, then that relation is transitive.
The identity and universal relations are transitive.

“is equal to”, “is greater than”, “is a subset of” these all are examples of transitive relation.

Show that the “is greater than” is a transitive relation for natural number?
Solution :
Let us consider three elements a, b and c of set “N” of natural numbers such that a relation “R” on “N” is :
(a, b)$\in$ R,(b, c)$\in$R,“is greater than”,a, b, c $\in$ N

This means that :
“a $\geq$ b” and “b $\geq$c”.
So naturally "a $\geq$ c".
i.e. aRb, bRc $\Rightarrow$ aRc.

Hence, we conclude that the relation "is greater than" is transitive relation.

Some examples based on reflexive relation, symmetric relation and transitive relation.

Example 1:
Determine the relation R in the set {a, b, c} given by
R = {(a, a), (b, b),(c, c), (a,b), (b, c)} is reflexive but neither symmetric nor transitive.
We have R = {(a, a), (b, b),(c, c), (a, b), (b, c)}. Here R is reflexive, because since
(a, a), (b, b) and (c, c) are in R.
R is not symmetric,as (a, b) $\in$ R but (b, a)$\notin $ R.
Again, R is not transitive, since
(a, b) $\in$ R and (b, c) $\in$ R but (a, c) $\notin $ R.

Example 2:
Let T be the set of all lines in a plane and R be the relation in T
defined as R = {(T1,T2) : T1 is perpendicular to T2}. Show that R is symmetric but neither reflexive nor transitive.
R is not reflexive, as a line T1 can not be perpendicular to itself, i.e.,
(T1, T1) $\notin $ R.
R is symmetric as (T1, T2) $\in$ R
$\Rightarrow$ T1 is perpendicular to T2
$\Rightarrow$ T2 is perpendicular to T1

$\Rightarrow$ (T2, T1) belongs to $\in$R.

R is not transitive. Because, if T1 is perpendicular to T2 and T2 is perpendicular to T3, then T1 can never be perpendicular to T3. In fact, T1 is parallel to T3, i.e.,
(T1, T2) $\in$ R, (T2, T3) $\in$ R but (T1, T3) $\notin $ R.
A relation R is said to be an equivalence relation if, it is reflexive, symmetric and transitive.
So we can say if,
  1. (a, b) $\in$ R, $\forall$ a, b ∈ A,
  2. (a, b) $\in$ R $\Rightarrow$ (b, a) $\in$ R, for all a, b ∈ A.
  3. (a, b) $\in$ R and (b, c) $\in$ R $\Rightarrow$ (a, c) $\in$ R, $\forall$ a, b, c $\in$ A.

Show that "=" is an equivalence relation.
To prove that "=" is an equivalence relation,
  1. simply a=a which is true for all a, hence "=" is reflexive.
  2. If a=b then b=a (obvious), so "=" is symmetry.
  3. If a=b and b=c so we can say a=c for all a, b and c, so "=" is transitive.
Hence "=" is an equivalence relation.