`"Empty Relation :"` 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 = φ ⊂ A xx A`.
`"Universal relation :"` 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 xx A`.
`=>` Both the empty relation and the universal relation are some times called trivial relations.
Eg.: Let A be the set of all students of a boys school. Show that the relation `R` in A given by `R = {(a, b)` : a is sister of b} is the empty relation and `R′ = {(a, b)` : the difference between heights of a and b is less than 3 meters} is the universal relation.
Solution: Since the school is boys school, no student of the school can be sister of any student of the school. Hence, `R = φ`, showing that `R` is the empty relation. It is also obvious that the difference between heights of any two students of the school has to be
less than `3` meters. This shows that `R′ = A xx A` is the universal relation.
`"(i) Reflexive Relation : "` : A relation in a set A is called reflexive relation if `(a,a) ∈ R` for every element `a ∈ A`. if `(a, a) ∈ R`, for every `a ∈ A`,
`"(ii) Symmetric Relation :"` A relation in a set `A` is called if `(a,b) ∈ R` the `(b,a) ∈ R` for all `a,b ∈ A`
if `(a_1, a_2) ∈ R` implies that `(a_2, a_1) ∈ R`, for all `a_1, a_2 ∈ A`.
`"(iii) Transitive Relation : "` A relation R on a set A is called transitive if whenever (a, b) is in R and (b, c) is in R, then (a, c) is in R. 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.`
`"Empty Relation :"` 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 = φ ⊂ A xx A`.
`"Universal relation :"` 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 xx A`.
`=>` Both the empty relation and the universal relation are some times called trivial relations.
Eg.: Let A be the set of all students of a boys school. Show that the relation `R` in A given by `R = {(a, b)` : a is sister of b} is the empty relation and `R′ = {(a, b)` : the difference between heights of a and b is less than 3 meters} is the universal relation.
Solution: Since the school is boys school, no student of the school can be sister of any student of the school. Hence, `R = φ`, showing that `R` is the empty relation. It is also obvious that the difference between heights of any two students of the school has to be
less than `3` meters. This shows that `R′ = A xx A` is the universal relation.
`"(i) Reflexive Relation : "` : A relation in a set A is called reflexive relation if `(a,a) ∈ R` for every element `a ∈ A`. if `(a, a) ∈ R`, for every `a ∈ A`,
`"(ii) Symmetric Relation :"` A relation in a set `A` is called if `(a,b) ∈ R` the `(b,a) ∈ R` for all `a,b ∈ A`
if `(a_1, a_2) ∈ R` implies that `(a_2, a_1) ∈ R`, for all `a_1, a_2 ∈ A`.
`"(iii) Transitive Relation : "` A relation R on a set A is called transitive if whenever (a, b) is in R and (b, c) is in R, then (a, c) is in R. 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.`