Question

Define a transitive relation.

Answer 1

Hint: The question is to discuss a transitive relation. Before discussing transitive relation we will discuss about relation, types of relations and then will discuss about transitive relation then we will take an example of relations and from that we will define or discuss about transitive relation.

Complete answer:
In the question, we have asked to define the transitive relation firstly we will define relation.
A relation is nothing but we can say that let \[A\] and \[B\] be non-empty sets then the relation \[R\] from \[A\] to \[B\] is defined as any subset of the Cartesian product $A \times B$ which mean we have to find out the Cartesian product $A \times B$ of $2$ non-empty sets \[A\] and \[B\] then subset of the Cartesian product is the relation $R$ from \[A\] to \[B\].
So, relation $R$ from \[A\] to be can be written as $R \subseteq A \times B$
Where $ \subseteq $ is the symbol for subset or equal to a subset of the Cartesian product of $A \times B$. There are three types of relations. Reflexives Relation, Symmetric relation and Transitive relation.
Therefore Transitive relation is defined as A relation $R$ on a set \[A\] is called transitive if $(a,b) \in R{\text{ and }}(b,c) \in R$ that implies $(b,c) \in R$ which means if $(a,b)$ the subset belongs to $R$ and $(b,c)$ is also a subset belonging to the subset $(a,c)$ must belong to $R$, this is the definition of transitive relation. Transitive relation is explained by taking as example of Relation $R$ as
$\Rightarrow$ $R = \left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right),\left( {1,2} \right),\left( {2,3} \right),\left( {1,3} \right)} \right\}$
Now, according to definition
If \[(1,2) \in R{\text{ and }}(2,3) \in R\]
Then \[(1,3) \in R{\text{ }}\]

That means the relation $R$ is transitive relation.

Note: If the relation is symmetric relation, reflexive relation and as well as is transitive relation. Then the relation is said to be the equivalence relation and the subset we are lacking from the Cartesian product$A \times B$, the subset can be the whole set too. This is the reason, we are using the sign $ \subseteq $ in$R \subseteq A \times B$.