In two different societies, there are some school going students inclu

1.	In two different societies, there are some school going students including girls as well as boys. Satish forms two sets with these students for his college project.Let = {a1, a2, a3, a4, a5 } and B= {b1, b2, b3, b4 } where ai's and bi's are school going students of first and second societies respectively.Satish decides to explore these sets for various types of relations and functions.With the help of above information answer the following question:Let R : A ⟶ A, R = {(x, y) : x and y are students of same set}. Then, discuss about reflexivity, symmetricity and transitivity of relation R.
Answer» We have relation R : A ⟶ A, R = {(x, y): x and y are students of same set}. ⇒ R = {(x, y) : x and y ∈ A}. Reflexivity : Let x ∈ A. ⇒ (x, x) ∈ R. Therefore, relation R is reflexive. Symmetricity : Let (x, y) ∈ R ⇒ x and y belongs to same set A. ⇒ y and x belongs to same set A ⇒ (y, x) ∈ R. Therefore, relation R is symmetric. Transitivity : Let (x, y) ∈ R and (y, z) ∈ R ⇒ x and y, y and z ∈ A ⇒ x and z ∈ A ⇒ (x, z) ∈ R. Therefore, relation R is transitive. Since, relation R is reflexive, symmetric and transitive. Therefore, relation R is an equivalence relation.

In two different societies, there are some school going students including girls as well as boys. Satish forms two sets with these students for his college project.Let = {a1, a2, a3, a4, a5 } and B= {b1, b2, b3, b4 } where ai's and bi's are school going students of first and second societies respectively.Satish decides to explore these sets for various types of relations and functions.With the help of above information answer the following question:Let R : A ⟶ A, R = {(x, y) : x and y are students of same set}. Then, discuss about reflexivity, symmetricity and transitivity of relation R.

Answer»

We have relation R : A ⟶ A, R = {(x, y): x and y are students of same set}.

⇒ R = {(x, y) : x and y ∈ A}.

Reflexivity : Let x ∈ A.

⇒ (x, x) ∈ R.

Therefore, relation R is reflexive.

Symmetricity : Let (x, y) ∈ R

⇒ x and y belongs to same set A.

⇒ y and x belongs to same set A

⇒ (y, x) ∈ R.

Therefore, relation R is symmetric.

Transitivity : Let (x, y) ∈ R and (y, z) ∈ R

⇒ x and y, y and z ∈ A

⇒ x and z ∈ A

⇒ (x, z) ∈ R.

Therefore, relation R is transitive.

Since, relation R is reflexive, symmetric and transitive.

Therefore, relation R is an equivalence relation.