Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}

Solutioin:

a ∈ A

|a - a| = 0 (which is even)

Therefore,

R is reflexive.

(a, b) ∈ R

⇒ |a - b| [is even]

⇒ |-(a - b)| = |b - a| [is even]

(b, a) ∈ R

Therefore,

R is symmetric.

(a, b) ∈ R and (b, c) Î R

⇒ |a - b| is even and |b - c| is even

⇒ (a - b) is even and (b - c )is even

⇒ (a - c) = (a + b) + (b - c) is even

⇒ |a - b| is even

⇒ (a, c) ∈ R

Therefore,

R is transitive.

R is an equivalence relation.

All elements of {1, 3, 5} are related to each other because they are all odd.

So, the modulus of the difference between any two elements is even.

Similarly, all elements {2, 4} are related to each other because they are all even.

No element of {1, 3, 5} is related to any elements of {2, 4} as all elements of {1, 3, 5} are odd and all elements of {2, 4} are even.

So, the modulus of the difference between the two elements will not be even

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.1 Question 8

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : a - b is even} is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Summary:

The relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation and also elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}