Question

If functions f: A → B and g: B → A satisfy gof = I_A, then show that f is one-one and g is onto

Answer 1

Given that, f: A → B and g: B → A satisfy gof = I_A

∵ gof =  I_A 

⇒ gof {f((x₁)} = gof {f(x₂)}

⇒ g(x₁) = g((x₂)  .......[∵ gof = I_A]

∴ x₁ = x₂

Hence, ‘f’ is also one-one and onto.