Question

Define the binary relation properties and prove that inverse relation R-1 is equivalence relation of R is equivalence relation

Answer 1

Please write in easy words

Answer 2

Binary relation property:

A binary relation of R is mainly defined on a set A and includes properties i.e. Reflexivity, Transitivity, Antisymmetry, Irreflexivity, Symmetry, and Asymmetry.

Prove:

Suppose R be a correspondence relation.

    Reflexive (a,a)   R   a A

Symmetric (a,b)   R

                 ⇒ (b,a)   R

Transitive (a,b), (b,c)   R

                ⇒ ac   R

Let R-1 is the converse relation of R

Here and now R-1 is reflexive

 (a,a)   R

⇒ (a,a)   R-1

Symmetric

(b,a)   R ⇒ (a,b)   R-1

(a,b)   R ⇒ (b,a)   R-1

(a,b), (b,a)   R-1

Consequently, R-1 is symmetric.

Transitive

(a,b), (b,c), (a,c)   R

Now, (b,a)   R-1 (a,c)   R-1

(c,b)   R-1

As R-1 is symmetric (b,c)   R-1

Therefore, R-1 is transitive.

So, R-1 is an equivalence relation.

Hope it helped...