Topic 4: Functions Flashcards
What is the definition of a function (2)
-Let A, B be sets
-A function f: A -> B is a rule that assigns to each element of A (the domain) a single element of B (the codomain)
What are some rules about the mapping of elements in functions (3)
-Every element of the domain A must have exactly one corresponding element of codomain B
-Not every element of B has to be mapped to by something in A
-Multiple elements of A can map to the same element in B
What is the image of x (2)
-The element y ∈ B mapped to by an element x ∈ A is called the image of x, often written y = f(x)
-y is a subset of B, consisting all the elements in A, called the image, image set or range of f
what is the function and image of f(x) = x^2 (2)
-f: ℝ -> ℝ; f(x) = x^2
-im(f) = ℝ0+ = [0, ∞)
What is the definition of a surjective function (3)
-A function f: A->B is surjective or onto if for every element in B, there exists an element in the domain that maps to it
-im(f) = B
-At least one element of A maps to all elements in B
What is the definition of an injective function (3)
-A function f: A->B is injective or one-to-one if f(a1) = f(a2) only if a1 = a2, for a1, a2 ∈ A
-f is injective if every element of B is mapped to by at most one element of A
-Not every element of B needs to be mapped to by an element of A, but no 2 x values can map to the same y value
What is the definition of a bijective function (2)
-A function f: A-> B is bijective if its injective and subjective
-if every element of A maps to exactly one element of B, and every element of B is mapped to by exactly one element of A
Whats one thing to note about whether a function is injective, surjective, bijective or neither, then x^2 as an example (1, 4)
-The domain and codomain of a function are fundamental aspects to a function
Take f = x^2
-f:ℝ -> ℝ is neither
-f: [0, ∞) -> ℝ is injective
-f: ℝ -> [0,∞) is surjective
-f: [0, ∞) -> [0, ∞) is bijective
How can we define composite functions (3)
-Suppose f: A -> B and g: C -> D, such that im(f) ⊆ C (all of B is within C)
-We can then chain the whole thing together (A -> D), call it the composite of f and g
-We can denote it (gof); A -> D such that (gof) (a) = g(f(a)) for all a ∈ A
How can we define inverse functions (1,3)
-Let X, Y be sets and suppose f: x -> y for some function.
-We say f is invertible if there exists some function f^(-1) : y -> x such that:
-(f^(-1) o f) (x) = x for all x ∈ X
-(f o f^(-1)) (y) = y for all y ∈ Y
What are 2 key propositions about functions being invertible (2)
-Function f: x -> y is invertible iff it is bijective (has to have each x value mapping to exactly one y value)
-If f: x -> y is invertible, then the inverse function f^(-1) y -> x is unique
