Question -
Answer -
(i) Given R = {(x, y): x and y work at the same place}
Now we have to check whether the relation is reflexive:
Let x be an arbitrary element of R.
Then, x ∈R
⇒ x and x work at the same place is true since they are the same.
⇒(x, x) ∈R [condition for reflexive relation]
So, R is a reflexive relation.
Now let us check Symmetric relation:
Let (x, y) ∈R
⇒x and y work at the same place [given]
⇒y and x work at the same place
⇒(y, x) ∈R
So, R is a symmetric relation.
Transitive relation:
Let (x, y) ∈R and (y, z) ∈R.
Then, x and y work at the same place. [Given]
y and z also work at the same place. [(y, z) ∈R]
⇒ 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.
Hence R is reflexive, symmetric and transitive.
(ii) Given R = {(x, y): x and y live in the same locality}
Now we have to check whether the relation R is reflexive, symmetric and transitive.
Let x be an arbitrary element of R.
Then, x ∈R
It is given that x and x live in the same locality is true since they are the same.
So, R is a reflexive relation.
Symmetry:
Let (x, y) ∈ R
⇒ x and y live in the same locality [given]
⇒ y and x live in the same locality
⇒ (y, x) ∈ R
So, R is a symmetric relation.
Transitivity:
Let (x, y) ∈R and (y, z) ∈R.
Then,
x and y live in the same locality and y and z live in the same locality
⇒ x, y and z all live in the same locality
⇒ x and z live in the same locality
⇒ (x, z) ∈ R
So, R is a transitive relation.
Hence R is reflexive, symmetric and transitive.
(iii) Given R = {(x, y): x is wife of y}
Now we have to check whether the relation R is reflexive, symmetric and transitive.
First let us check whether the relation is reflexive:
Let x be an element of R.
Then, x is wife of x cannot be true.
⇒ (x, x) ∉R
So, R is not a reflexive relation.
Symmetric relation:
Let (x, y) ∈R
⇒ x is wife of y
⇒ x is female and y is male
⇒ y cannot be wife of x as y is husband of x
⇒ (y, x) ∉R
So, R is not a symmetric relation.
Transitive relation:
Let (x, y) ∈R, but (y, z) ∉R
Since x is wife of y, but y cannot be the wife of z, y is husband of x.
⇒ x is not the wife of z
⇒(x, z) ∈R
So, R is a transitive relation.
(iv) Given R = {(x, y): x is father of y}
Now we have to check whether the relation R is reflexive, symmetric and transitive.
Reflexivity:
Let x be an arbitrary element of R.
Then, x is father of x cannot be true since no one can be father of himself.
So, R is not a reflexive relation.
Symmetry:
Let (x, y) ∈R
⇒ x is father of y
⇒ y is son/daughter of x
⇒ (y, x) ∉R
So, R is not a symmetric relation.
Transitivity:
Let (x, y) ∈R and (y, z) ∈R.
Then, x is father of y and y is father of z
⇒ x is grandfather of z
⇒ (x, z) ∉R
So, R is not a transitive relation.