# Exercise 1.1

Question 1

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

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

Sol :

(Reflexivity)

Let x be an arbitrary element of R . Then,

x and x work at the same place is true since they are same

So, R is a reflexive relation

(Symmetry)

Let

x and y work at the same place .

y and x work at the same place .

So, R is symmetric relation

(Transitivity)

Let and . Then,

x and y work at the same place .

y and z work at the same place .

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

So, R is a transitive relation.

(ii) R = {(x , y) : x and y live in the same locality }

Sol :

(Reflexivity)

Let x be an arbitrary element of R . Then,

x and x live in the same locality is true since they are same.

So, R is a reflexive relation

(Symmetry)

Let

x and y live in the same locality.

y and x live in the same locality.

So, R is symmetric relation

(Transitivity)

Let and . Then,

x and y live in the same locality.

y and z live in the same locality.

x , y and z live in the same locality.

So, R is a transitive relation.

(iii) R = {(x , y) : x is wife of y }

Sol :

(Reflexivity)

Let x be an element of R . Then,

x is wife of x which cannot be true.

So, R is not a reflexive relation

(Symmetry)

Let

x is wife of y.

x is female and y is male.

y cannot be wife of x as y is husband of x.

So, R is not a symmetric relation.

(Transitivity)

Let and . Then,

x is wife of y and y is husband of z

which is a contradiction

So, R is not transitive

(iv) R = {(x , y) : x is father of and y }

Sol :

(Reflexivity)

Let x be an arbitrary element of R . Then,

x is father of x which cannot be true since no one can be father of himself.

So, R is not a reflexive relation

(Symmetry)

Let

x is father of y.

y is son or daughter of x .

So, R is not a symmetric relation.

(Transitivity)

Let and . Then,

x is father of y and y is father of z

x is grandfather of z.

So, R is not transitive

Question 2

Three relations are defined on a set as follows:

={(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b),(c,c)}

={(a,b),(b,c),(c,a)}

Find whether or not each of the relations on A is (i) reflexive (ii) symmetric (iii) transitive.

Sol :

(i) ={(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b),(c,c)}

Reflexive as , and

Not symmetric as but

Not transitive as and but

(ii)

Not Reflexive as but and

Symmetric as and

Transitive as and also

(iii)

Not Reflexive as , and

Not symmetric as but

Transitive as has only two elements

(iv) ={(a,b),(b,c),(c,a)}

Not Reflexive as , and

Not symmetric as but

Not transitive as and but

Question 3

Test whether the following relations are (i) reflexive (ii) symmetric and (iii) transitive

(i) defined by

Sol :

(Reflexivity)

Let “a” be an arbitrary element of .Then,

for all

So, is not reflexive.

(Symmetry)

Let . Then,

So, is symmetric.

(Transitivity)

Let and

and

So, is not transitive

(ii) defined by

Sol :

(Reflexivity)

Let “a” be an arbitrary element of .Then,

So, is reflexive.

(Symmetry)

Let . Then,

So, is symmetric.

(Transitivity)

Let and

So, is not transitive

For better illustration example is give

but

(iii) defined by

Sol :

(Reflexivity)

Let “a” be an arbitrary element of .Then,

So, is reflexive.

(Symmetry)

Let . Then,

for all

So, is not symmetric.

(Transitivity)

Let and

and

and

So, is not transitive

Question 4

Let A={1,2,3} and let ={(1,1),(1,3),(3,1),(2,2),(2,1),(3,3)}

={(2,2),(3,1),(1,3)}

={(1,3),(33)}

Find whether or not each of the relations on A is (i) reflexive (ii) symmetry (iii) transitive

Sol :

(i) ={(1,1),(1,3),(3,1),(2,2),(2,1),(3,3)}

Reflexive as , and

Symmetric as but

Not Transitive as and but

(ii) ={(2,2),(3,1),(1,3)}

Not Reflexive as and

Symmetric as but

Not Transitive as and but

(iii) ={(1,3),(33)}

Not Reflexive as and

Not Symmetric as but

Transitive as and also

Question 5

The following relations are defined on the set of real numbers .

(i) [latex]aRb \text{ if } a-b>0[latex]

(ii) [latex]aRb \text{ if } 1+ab>0[latex]

(iii) [latex]aRb \text{ if } |a|\leq b[latex]

Sol :

(i) [latex]aRb \text{ if } a-b>0[latex]

( Reflexivity )