Question

What is transitive relation?

Answer 1

Hint : By relation, we understand a connection or link between the two people, or between the two things. In the set theory, a relation is a way of showing a connection or relationship between any two sets. There are different types of relations, namely empty, universal, identity, inverse, reflexive, symmetric and transitive relation.

Complete step-by-step answer :
Now, we are going to define what transitive relation is.
A transitive relation on set:
Let A be any set defined on the relation R. then R is said to be a transitive relation if
$(a,b) \in R$ and $(b,c) \in R$ $\Rightarrow (a,c) \in R$ .
That is aRb and bRc $\Rightarrow$ aRc where $a,b,c \in R.$
[We can also define a non-transitive relation. That is the relation is said to be non-transitive, if
$(a,b) \in R$ and $(b,c) \in R$ does not implies $(a,c) \in R$ ]
For example:
Let R= {(a, b): $(a,b) \in z$ and $(a - b)$ is divisible by K.}
$\Rightarrow$ Let $a,b,c \in R$ .
$\Rightarrow$ Assume $(a,b) \in R$ and $\Rightarrow (b,c) \in R$
$\Rightarrow$ Now (a-b) is divisible by K and (b-c) is divisible by K ( $\because$ of function defined in R)
$\Rightarrow$ $\\{ a - b + b - c\\}$ is divisible by K.
$\Rightarrow$ (a-c) is divisible by K
$\Rightarrow (a,c) \in R$ ( $\because$ of function defined in R)
Hence, from above we have, $(a,b) \in R$ and $(b,c) \in R$ $\Rightarrow (a,c) \in R$ .
Hence, R is a transitive relation.
So, the correct answer is “ $(a,b) \in R$ and $(b,c) \in R$ $\Rightarrow (a,c) \in R$ THEN “R”IS SAID TO BE TRANSITIVE. ”.

Note : Equivalence relation R in A is a relation which is reflexive, symmetric and transitive. Equivalence relation is also called a bijective function. Reflexive function R in A is a relation with $(a,a) \in R$ . Symmetric relation R in X is a relation satisfying $(a,b) \in R$ $\Rightarrow (b,a) \in R$ . In the symmetric, the second element of the first pair is the same as the first element of the second pair ( $(a,b)$ , $(b,c)$ ). If not then it is not a transitive relation.