Question

What are transitive relations?

Answer 1

Define transitive relations.

Relation means two objects or quantities or elements of a set are related if there is a recognizable connection or link between the two objects or quantities or elements.

Example: If $(a,b)\in R$, we say that $a$ is related to $b$ under the relation $\mathrm{R}$ where $a$and $b$are objects or quantities or elements of a set

Transitive Relation:

A relation $\mathrm{R}$ in a set$\mathrm{A}$ is called transitive, if $(x,y)\in R,(y,z)\in R$ then$(x,z)\in R$, for all$x,y\mathrm{and}z\in \mathrm{A}$.

For example: Let $A=\left\{a,b,c,d\right\}$ and $\mathrm{R}$ be a relation on $\mathrm{A}$ in such way that

for$a$ and$b$,$\left\{\left(\mathrm{a},\mathrm{b}\right)\in \mathrm{R}:a\mathrm{and}b\mathit{}\mathrm{are}\mathrm{brothers}\right\}$

also for $b$and$c$,$\left\{\left(b,c\right)\in \mathrm{R}:b\mathrm{and}c\mathit{}\mathrm{are}\mathrm{brothers}\right\}$

then this relation $\mathrm{R}$ must exist for $a$and $c$ $\left\{\left(a,c\right)\in \mathrm{R}:a\mathrm{and}c\mathit{}\mathrm{are}\mathrm{brothers}\right\}$ here the relation is transitive.

Hence, For a transitive relation in the set$\mathrm{A}$ ,If$(x,y)\in R,(y,z)\in R$, then$(x,z)\in R$ where $x,y\mathrm{and}z\in \mathrm{A}$