`\color{purple} "★ DEFINITION ALERT"`
`\color{fuchsia} \mathbf\ul"TRANSITIVE RELATION"`
`\color{purple} ✍️` A relation `R` in a set `A` is called `color{green}ul"transitive"`, if `(a_1, a_2) ∈ R` and `(a_2, a_3)∈ R` implies that `(a_1, a_3)∈ R,` for all `a_1, a_2, a_3 ∈ A.`
`\color{purple} ✍️` Let `A` be a set in which the relation `R` defined.
`R` is said to be transitive, if
`(a, b) ∈ R` and `(b, a) ∈ R ⇒ (a, c) ∈ R,`
That is `a \ \R\ \b` and `b\ \ R\ \ c ⇒ a \ \ R\ \ c` where `a, b, c ∈ A.`
`\color{purple} ✍️` The relation is said to be `color{green}ul"non-transitive"`, if
`(a, b) ∈ R` and `(b, c) ∈ R` do not imply `(a, c ) ∈ R.`
`color{green}"Example -"` `A={ 0,1,2,3}` and a relation `R` on `A` be given by
`R={ (0,0), (0,1), (0,3), (1,0), (1,1), (2,2), (3,0), (3,3) }`
`R` is not transitive because `(1,0) ∈ R, \ \ (0,3) ∈ R` and `(1,3) ∉R` imply `R` is not transitive.
`\color{purple} "★ DEFINITION ALERT"`
`\color{fuchsia} \mathbf\ul"TRANSITIVE RELATION"`
`\color{purple} ✍️` A relation `R` in a set `A` is called `color{green}ul"transitive"`, if `(a_1, a_2) ∈ R` and `(a_2, a_3)∈ R` implies that `(a_1, a_3)∈ R,` for all `a_1, a_2, a_3 ∈ A.`
`\color{purple} ✍️` Let `A` be a set in which the relation `R` defined.
`R` is said to be transitive, if
`(a, b) ∈ R` and `(b, a) ∈ R ⇒ (a, c) ∈ R,`
That is `a \ \R\ \b` and `b\ \ R\ \ c ⇒ a \ \ R\ \ c` where `a, b, c ∈ A.`
`\color{purple} ✍️` The relation is said to be `color{green}ul"non-transitive"`, if
`(a, b) ∈ R` and `(b, c) ∈ R` do not imply `(a, c ) ∈ R.`
`color{green}"Example -"` `A={ 0,1,2,3}` and a relation `R` on `A` be given by
`R={ (0,0), (0,1), (0,3), (1,0), (1,1), (2,2), (3,0), (3,3) }`
`R` is not transitive because `(1,0) ∈ R, \ \ (0,3) ∈ R` and `(1,3) ∉R` imply `R` is not transitive.