Question

16. Let be the equivalence relation defined on = {1, 2, 3} by

= {(1,1), (2,2), (3,3), (1,3), (3,1)} , where [] is the equivalence

class of

(a) Find [1], [2], [3] (1)

(b) (i) Identify the relationship between [2] and [3] (1)

(ii) Find [1] ∩ [2] (1)

(iii) Find [1] ∪ [2]​

Answer 1

The relation R on set A={1,2,3,4,5,6,7} is defined by

R={(a,b): both a and b are either odd or even}

We observe the following properties of R on A

Reflexivity: Clearly, (1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7)∈R. So, R is a reflexive relation in A

Symmetric: Let a,b∈A be such that (a,b)∈R.

Then, (a,b)∈R

Both a and b are either odd or even

Both b and a are either odd or even

⇒ (b,a)∈R

Thus, (a,b)∈R⇒(b,a)∈R for all a,b∈A

So, R is a symmetric relation on A

Transitivity: Let a,b,c∈Z be such that (a,b)∈R,(b,c)∈R.

Then, (a,b)∈R⇒ Both a and b are either odd or even

(b,c)∈R⇒ Both b and c are either odd or even

If both a and b are even, then

(b,c)∈R⇒ Both b and c are even

If both a and b are odd, then

(b,c)∈R⇒ Both b and c are odd

∴ Both a and c are even or odd. Therefore (a,c)∈R

So, (a,b)∈R and (b,c)∈R⇒(a,c)∈R

Consequently, R is a transitive relation on A

Hence, R is an equivalence relation on A

We observe that two numbers in A are related if both are odd or both are even.

Since {1,3,5,7} has all odd numbers of A. So, all the numbers of {1,3,5,7} are related to each other.

Similarly, all the numbers of {2,4,6} are related to each other as it contains all even numbers of set A.

An even ,odd number in A is related to an even ,odd number in A respectively.

So, no number of the subset {1,3,5,7} is related to any number of the subset {2,4,6}