Functions

Question 1

What is a function?

Answer 1

A rule which associates to every element a ∈ A, one and only one element f(a) in B.

We write f:A -> B to indicate that f is a function from A to B where Set A is the Domain and B is the Codomain

Question 2

Let f : A → B and g : C → D be functions. When are f and g equal?

Answer 2

if The domains and codomains are equal

Question 3

What is a surjective function

Answer 3

We say f is surjective (or f is onto, or f is a surjection) if, for
each b ∈ B, there is at least one a ∈ A with f(a) = b.

(Surjective means that every “B” has at least one matching “A” (maybe more than one)

Question 4

What is an injective function?

Answer 4

We say f is injective (or f is one-to-one, or f is an injection)
if, for each b ∈ B, there is at most one a ∈ A with f(a) = b. So
to say that f is injective means that, for any elements a, a
′ ∈ A,

(1 to 1)

Question 5

What is an identity function

Answer 5

idA: a -> a
for all a in A

f ◦ idA = f and idB ◦ f = f .

Question 6

what is an inverse function

Answer 6

An inverse (function) to a function f : A → B is a function g: B → A such that
g ◦ f = idA and f ◦ g = idB.

Question 7

When can a function have an inverse function

Answer 7

A function f : A → B has an inverse function if and only if f is a bijection.

Question 8

what is The inverse of a composite of bijections

Answer 8

Let f : A → B and g : B → C be bijections. Then g ◦ f : A → C is also a bijection, and its inverse is

then ((g◦ f)−1 = f−1 ◦g−1.)

Question 9

What is a one sided inverse?

Answer 9

for a function F: a -> b,
i) g: B → A is a left inverse to f if g ◦ f = idA;
(ii) h: B → A is a right inverse to f if f ◦ h = idB.

One-sided inverses (i.e. left inverses or right inverses) need not be unique, and a function can have a one-sided inverse without being a bijection.

Question 10

What is a permutation

Answer 10

Let X be any set. Then a permutation on X is a bijection π: X → X.

Question 11

What are the properties of permutations

Answer 11

• If σ, τ are permutations on X, so is σ◦τ.
• For any permutations σ, τ, π on X, the permutations (σ ◦ τ) ◦ π and σ ◦ (τ ◦ π) are equal.
• The identity function idX : X → X is a permutation on X.
• Each permutation σ on X has an inverse function σ−1 : X → X, so that σ◦σ−1 = idX and σ−1 ◦σ = idX. The function σ−1 is itself a permutation. (It is a bijection since it has an inverse function, namely σ itself.)