Functions

Question 1

Whats the formal definition of a function?

Answer 1

A relation f is a function provided (a, b) in f and (a, c) in f imply b = c.

Question 2

Whats the formal definition of function notation?

Answer 2

Let f be a function and let a be an object, the notation f(a) is defined provided there exists an object b such that (a, b) in f. In this case f(a) = b

Question 3

Bijection

Answer 3

A function f:A->B is a bijection IFF

1. Injection (one-to-one)
   for all x1, x2, in A, if f(x1)=f(x2), then x1=x2. ( maps distinct elements of its domain to distinct elements of its codomain)
2. Surjection (onto)
   for all b in B, there exists an a in A such that f(a) = b. (The image of f spans the entire codomain)