Chapter 12 Functions

Define an injective function

A function f: A→B is injective if for every x,y∈A, x≠y implies f(x)≠f(y). It means the function is one to one. In essence, injective means that unequal elements in A always get sent to unequal elements in B

Define a surjective function

A function f: A→B is surjective if for every b∈B there is an a∈A with f(a)=b (pg. 201) Surjective means that every element of B has an arrow pointing to it, that is, it equals f (a) for some a in the domain of f Note: a function is surjective if and only if its codomain equals its range (pg. 202)

Define a bijective function

A function is bijective if it is both injective and surjective (pg. 201).

Formal definition of relation from a set A to another set B.

A relation from a set A to a set B is a subset R⊆AxB. We often abbreviate the statement (x,y)∈R as xRy.

Is composition of functions associative? Commutative?

Composition of functions is associative. So (h◦ g) ◦ f = h◦ (g ◦ f ) Composition of functions is NOT commutative so g◦ f does not necesarily equal f ◦ g (pg. 209)

Define domain, codomain, and range.

For a function f:A→B, the set A is called the DOMAIN of f. (Think of domain as the set of possible "input values" for f.) The set B is called the codomain of f. The range of f is the set {f(a):a∈A}={b: (a,b)∈f}. (Think of the range as the set of all possible "output values" for f. Think of the codomain as a sort of "target" for the outputs.)(pg. 199)

Given two injective functions, what do we known about their composition? Prove it. Given two surjective functions, what do we know about their composition? Prove it.

Given two injective functions, their composition is injective. Given two surjective functions, their composition is surjective.

Is the function f(x)=x^2 (from ℝ to ℝ) injective? Surjective? Bijective?

It's not injective because -2≠2, but f(-2)=f(2). It's not surjective, because if b=-1 (or any negative), then there is no a∈ℝ with f(a)=b. So it's not bijective either (pg. 202)

What is an identity function?

It's outputs are it's inputs. e.g. y=x

Define an inverse relation

Out outputs become our inputs and our inputs become our outputs.

Formal definition of a function.

Suppose A and B are sets. A function f from A to B (denoted as f: A→B) is a relation f⊆AxB from A to B, satisfying the property that for each a∈A the relation contains exactly one ordered pair of form (a,b). The statement (a,b)∈f is abbreviated f(a)=b. (pg. 197)

How do we show a function f: A→B is surjective?

Suppose b∈B and prove there exists a∈A for which f(a)=b

Differentiate between domain, codomain, and range.

The domain is everything that can go into the function. The codomain is everything that could possibly come out of the function. The range is everything that DOES come out of the function.

Formal definition of equality of functions

Two function f: A→B and g: A→D are equal if f(x)=g(x) for every x∈A. Observe that f and g can have different codomains and still be equal!!! (pg. 200)

How do we show a function f: A→B is injective?

We can do so directly: suppose x,y∈A and x≠y. .... Therefore f(x)≠f(y). We can also take the contrapositive approach: suppose x,y∈A and f(x)=f(y). .... Therefore x=y (pg. 203)

How do we prove a function f is NOT surjective?

We must prove the negation of the statement ∀b∈B, ∃a∈A, f(a)=b, that is, we must prove ∃b∈B,∀a∈A, f(a)≠b (pg. 203)

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f−1(f (X)) is a subset of what?

Y

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f (W ∪ X) equals what?

f (W ∪ X) = f (W)∪ f (X)

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f (W ∩ X) is a subset of what?

f (W)∩ f (X)

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f−1(Y ∩ Z) equals what?

f−1(Y)∩ f−1(Z)

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f−1(Y ∪ Z) equals what?

f−1(Y)∪f−1(Z)

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. X is a subset of what?

f−1(f (X))

Define the composition of two functions

g∘f=g(f(x)) f∘g=f(g(x))

Prove that composition of functions is associative

pg. 209

Define image and preimage

pg. 215

Summarize the four different injective/surjective combinations that a function may posses.

see the pic