8 - Injections, Surjections, and Bijections

Question 1

Definition 8.1 (Injective)

When is a function said to be injective?

Answer 1

A function f : A→B is injective if

∀x,y∈A, x≠y ⇒ f(x)≠f(y)

Question 2

Eq. Definition 8.1 (Injective)

When is a function equivalently said to be injective?

Answer 2

A function is equivalently said to be injective if

∀x,y∈A, f(x)=f(y) ⇒ x=y

Question 3

How do you prove something is not injective?

Answer 3

It’s enough to find two elements x and y of the domain such that x≠y and f(x)=f(y) (negation)

Question 4

Definition 8.4 (Surjective, surjection, image)

When is a function said to be surjective?

Answer 4

A function f : A→B is surjective (or is a surjection onto, or is onto) if
∀b∈B ∃a∈A such that f(a)=b

Question 5

What is the image of S under f for any subset S⊆A?

Answer 5

∀S⊆A, the image of S under f is the set

f(S) := {f(a) : a∈S} ⊆ B

Question 6

What is the image of f and when is f surjective?

Answer 6

The image of f is f(A). In other words, the image of f is the image of the domain of f under f. Thus, f is surjective if and only if f(A)=B (i.e. the image of f is equal to the codomain of f)

Question 7

Prove that the function
f: Z - {0} → Z>0, f(x)=|x|
is surjective

Answer 7

To prove this, note that

∀x∈Z>0, we have f(x)=|x|=x

Question 8

Definition 8.1 (Bijective)

When is a function said to be bijective?

Answer 8

A function is bijective (or is a bjection) if it is both injective and surjective.