Question

Let A be a set with |A| = n, and let R be an equivalence relation on A with |R| = r. Why is r − n always even?

Answer 1

Answer:

Correct option is C)

Let R be a relation on A i.e. R⊆A×A

R={(a,b)∣a,b∈A}

Also, given R is equivalence relation,

Now, let R

−1

={(b,a)∣(a,b)∈R}

We will check whether R

−1

is reflexive, symmetric, transitive or an equivalence relation.

Reflexive:

Since, R is reflexive

⇒(a,a)∈R

⇒(a,a)∈R

−1

(by def of R

−1

)

Hence, R

−1

is reflexive.

Symmetric: Let (b,a)∈R

−1

⇒(a,b)∈R (by def of R

−1

)

⇒(b,a)∈R (Since, R is symmetric)

⇒(a,b)∈R

−1

(by def of R

−1

)

Hence, R

−1

is symmetric.

Transitive : Let (b,a),(a,c)∈R

−1

⇒(a,b),(c,a)∈R (by def of R

−1

)

or (c,a)(a,b)∈R

⇒(c,b)∈R (since, R is transitive.)

⇒(b,c)∈R

−1

Hence, R

−1

is transitive.

Hence, R

−1

is an equivalence relation.