`\color{green} ✍️` The concept of the term `color{green} "‘relation’"` in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities.

`\color{green} ✍️` Let `A` be the set of students of Class `XII` of a school and `B` be the set of students of Class `XI` of the same school.

Then some of the examples of relations from `A` to `B` are
`(i) {(a, b) ∈ A × B: "a is brother of b" },`
`(ii) {(a, b) ∈ A × B: "a is sister of b"},`
`(iii) {(a, b) ∈ A × B: "age of a is greater than age of b"},`

If `(a, b) ∈ R,` we say that `a` is related to `b` under the relation `R` and we write as `a \ \ R \ \ b.` In general, `(a, b) ∈ R,` we do not bother whether there is a recognisable connection or link between `a` and `b.` As seen in Class `XI,` functions are special kind of relations.

`\color{green} ✍️` we would like to study different types of relations. We know that a relation in a set A is a subset of `A × A.`

`\color{purple} "★ DEFINITION ALERT"`

`\color{fuchsia} \mathbf\ul"EMPTY RELATION"`

`\color{purple} ✍️` A relation R in a set `A` is called empty relation, if no element of `A` is related to any element of `A`, i.e., `R = phi ⊂ A × A.`

`\color{purple} ✍️` For illustration, consider a relation `R` in the set `A = {1, 2, 3, 4}` given by `R = {(a, b): a – b = 10}.` This is the empty set, as no pair `(a, b)` satisfies the condition `a – b = 10.`

`\color{purple} "★ DEFINITION ALERT"`

`\color{fuchsia} \mathbf\ul"UNIVERSAL RELATION"`

`\color{purple} ✍️` A relation `R` in a set `A` is called universal relation, if each element of `A` is related to every element of `A`, i.e., `R = A × A.`

Both the empty relation and the universal relation are some times called trivial relations.

`\color{green} ★ \color{green} \mathbf(KEY \ CONCEPT)`
`\color{navy} "If (a, b) ∈ R, we say that a is related to b and we denote it as"` `a \ \R \ \b.`

`\color{purple} "★ DEFINITION ALERT"`

`\color{fuchsia} \mathbf\ul"REFLEXIVE RELATION"`

`\color{purple} ✍️` A relation `R` in a set `A` is called `color{green}ul"reflexive"` if `(a, a) ∈ R,` for every `a∈ A,`

`\color{purple} ✍️` It means `R` is `color{green}ul"reflexive"` if every element of `a∈ A` is related to itself.

`\color{green} ✍️` A relation R in a set A is `color{green}ul"not reflexive"` if there be at least one element `a ∈ A` such that `(a, a) ∉ R.`

`color{green}"Example -"` a set `A = {p, q, r, s}.`

`\color{green} ✍️` The relation `R_1 = {(p, p), (p, r), (q, q), (r, r), (r, s), (s, s)}` in `A` is reflexive,

since every element in `A` is `R_1`-related to itself.

`\color{green} ✍️` But the relation `R_2 = {(p, p), (p, r), (q, r), (q, s), (r, s)}` is not reflexive in `A` .

since `q, r, s ∈ A` but `(q, q) ∉ R_2, (r, r) ∉ R_2` and `(s, s) ∉ R_2.`

`\color{purple} "★ DEFINITION ALERT"`

`\color{fuchsia} \mathbf\ul"SYMMETRIC RELATION"`

`\color{purple} ✍️` A relation `R` in a set `A` is called `color{green}ul"symmetric"`, if `(a_1, a_2) ∈ R` implies that `(a_2, a_1)∈ R,` for all `a_1, a_2 ∈ A.`

`\color{purple} ✍️` It means if `a` is related to `b` then `b` also related to `a`.

`color{green}"Example -"` a set `A = {1, 2, 3, 4}.`

`\color{green} ✍️` The relation `R_1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4) }`

`R_1` is not symmetric because `(3, 4) in R_1 ` and `(4, 3) ∉ R_1 `

`\color{green} ✍️` But the relation `R_2 ={(1, 1), (1, 2), (2, 1)}`

`R_1` is symmetric because `(1, 2) in R_1 ` and `(2, 1) ∉ R_1 `

`\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"EQUIVALENCE RELATION"`

`\color{purple} ✍️` A relation `R` in a set `A` is said to be an `color{green}ul"equivalence relation"` if `R` is reflexive, symmetric and transitive.

`\color{purple} ✍️` A relation `R` in a set `A` is called
`"(i) reflexive,"` if `(a, a) ∈ R,` for every `a∈ A,`

`"(ii) symmetric,"` if `(a_1, a_2) ∈ R` implies that `(a_2, a_1)∈ R,` for all `a_1, a_2 ∈ A.`

`"(iii) 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{green}"Example -"` `A={1,2,3,4}` and a relation `R` on `A` be given by

`R={ (1,1), (1,2), (2,1), (2,2), (3,4), (4,3), (3,3), (4,4) }`

then `R` is a equivalence relation.