composition of injective and surjective functions

The composition of g with f, denoted g ◦ f, is the function from A to C defined by (g ◦ f)(x) = g(f(x)) The composition of two functions is defined by following one function by another To define the composition g ◦f we must have the range of f contained in the domain of g

The composition of two injective functions is injective Proofs 1 Suppose f: A→B and g: B→C are surjective (onto) To prove that gοf: A→C is surjective, we need to prove that ∀c∈C ∃ a∈A such that (gοf)(a) = c

the formal definitions of injectivity and function composition Page 16 Theorem: If f : A → B is an injection and g : B → C 

23 fév 2009 · Now, we need to apply the definition of function composition and the fact that f and g are each injective: Proof: Let A, B, and C be sets Let f : A → 

Definition 0 5 Suppose that f : B → C is one function and g : A → B is another function One can form the composition f ◦ g 

If A and B are finite sets, an arrow diagram shows a function f Three properties: surjective (onto), injective, bijective composition function, , is a function

i) Function f is injective iff f−1({b}) has at most one element for all b ∈ B ii) Function f is surjective iff and bijectivity is preserved by composition of functions 1 

one-to-one and onto (or injective and surjective), how to compose functions, and when Then the composition of f and g is a new function denoted by g ◦ f, and

Conclude that f is not injective Let's consider how injectivity plays with function composition Theorem 2 Let X,Y,Z be sets, and let f : X 

28 oct 2011 · (b) Show that if g ◦ f is surjective then g is surjective Solution First, we prove (a) Suppose that g◦f is injective; we show that f is injective To this