Functions Flashcards
function
maps elements of a set X to elements of a set Y, is a subset of X x Y such that for every x ∈ X, there is exactly one y ∈ Y for which (x,y) ∈ f
domain (chapter 5)
the set X of f
target
the set Y of f
f maps x to y can be denoted as…
(x, y) ∈ f OR f(x) = y
when is f well-defined
when one element of the domain is mapped to one element of the target
when is an element y in the range of f?
if and only if there is an x ∈ X such that (x,y) ∈ f
Expressed in set notation as:
Range of f = { y: (x, y) ∈ f, for some x ∈ X }
when are two functions equal?
if f and g have the same domain and target, and f(x) = g(x) for every element in the domain
floor function
maps real number to nearest integer in the downward direction
ceiling function
maps real number to nearest integer in the upward direction
nested floor and ceiling functions
perform operation on inside first (like brackets)
one-to-one (injective)
if x1 != x2, f maps different elements in X to different elements in Y
onto (surjective)
if the range of f is equal to the target Y, for every y ∈ Y, there is an x ∈ X such that f(x) = y
bijective
if it is both one-to-one and onto
if function is onto
|D| >= |T|
if function is one-to-one
|D| <= |T|
if function is bijection
|D| <= |T| and |D| >= |T|
inverse
f is obtained by exchanging the first and second entries
f = (x,y), f^-1 = (y,x), if bijection
when does a function have an inverse?
f: X → Y has an inverse if and only if reversing each pair in f results in a well-defined function from Y to X
composition
the process of applying a function to the result of another function
identity function
always maps a set onto itself and maps every element onto itself eg. IvA: A → A, Iva(a) = a for all a ∈ A
