Let f: W → W be defined as. Then show that f is invertible. Also, find the inverse of f.

OR

Show that the relation R in the set N×N defined by (a,b)R(c,d) iff a²+d²= b²+ c² for all a, b, c, d є N, is an equivalence relation.

Consider the case when n is odd then n–1 should be even

Using the definition

f: W → W is such that

f(f(n))=f(n–1)=n–1+1=n

Consider the case when n is even then n+1 should be odd

Using the definition

fof: W → W is such that

f(f(n))=f(n+1)=n+1–1=n

Therefore fof = I

So f is invertible and f = f^–1.

OR

To prove R is an equivalence relation we need to prove that R is reflexive, symmetric and transitive

Let (x, y) є N × N

So, x²+y²= y²+ x²

Here (x, y)R(x, y)

Therefore R is reflexive.

Let (x, y), (s, t) є N × N such that (x, y)R(s, t)

So x²+t²= y²+ s²

⇒ s²+y²= t²+ x²

⇒ (s, t)R(x, y)

Therefore R is symmetric

Let (x, y), (s, t), (m, n) є N × N such that (x, y)R(s, t) and (s, t)R(m, n)

⇒ x²+t² = y²+s² and s²+n²= t²+m²

Adding both

x²+t²+s²+n²= y²+ s²+t²+ m²

⇒ x²+n²= y²+ m²

⇒ (x, y)R(m, n)

Therefore R is transitive.

R is reflexive, symmetric and transitive and thus an equivalence relation.