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.