Question

Let A be the set of all human beings in a town at a particular time. Determine whether of the following relation is reflexive, symmetric and transitive :

 R = {(x, y) : x and y work at the same place}

determination of whether the following relations are reflexive, symmetric, and transitive:

Relation R in the set A of human beings in a town at a particular time given by R = {(x, y) : x and y work at the same place}

Answer 1

(i) Reflexivity: 

Let x be an arbitrary element of R. Then,

x∈R   

⇒x and x work at the same place, which is true since they are the same.

⇒(x, x) ∈R

So, R is a reflexive relation.

(ii) Symmetry :

Let (x, y)∈R

⇒x and y work at the same place 

⇒y and x work at the same place

⇒(y, x)∈R

So, R is a symmetric relation.

(iii) Transitivity:

Let (x, y)∈R and (y, z)∈R. Then,

x and y work at the same place.

y and z also work at the same place.

⇒ x , y and z all work at the same place.

⇒x and z work at the same place.

⇒ (x, z)∈R

So, R is a transitive relation.