If f : A → B and g : B → C are two functions such that g o f : A → C is a bijection, then which one of the following is always true?

Step-by-step solution

Given: f : A → B and g : B → C And, g o f : A → C is a bijection. Let x ,   y ∈ A such that f x = f y . Then, f x = f y ⇒ g f x = g f y ⇒ g o f x = g o f y ⇒ x = y ;       ∵ g o f : A → C  is one-one Hence, f x is an injection. Now, Let z be an arbitrary element of C . Since, g o f : A → C is onto. Therefore, there exists x ∈ A such that g o f x = z ⇒ g f x = z ⇒ g y = z where y = f x ∈ B . Thus, for al