Functions

Question 1

function

Answer 1

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

Question 2

domain (chapter 5)

Answer 2

the set X of f

Question 3

target

Answer 3

the set Y of f

Question 4

f maps x to y can be denoted as…

Answer 4

(x, y) ∈ f OR f(x) = y

Question 5

when is f well-defined

Answer 5

when one element of the domain is mapped to one element of the target

Question 6

when is an element y in the range of f?

Answer 6

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 }

Question 7

when are two functions equal?

Answer 7

if f and g have the same domain and target, and f(x) = g(x) for every element in the domain

Question 8

floor function

Answer 8

maps real number to nearest integer in the downward direction

Question 9

ceiling function

Answer 9

maps real number to nearest integer in the upward direction

Question 10

nested floor and ceiling functions

Answer 10

perform operation on inside first (like brackets)

Question 11

one-to-one (injective)

Answer 11

if x1 != x2, f maps different elements in X to different elements in Y

Question 12

onto (surjective)

Answer 12

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

Question 13

bijective

Answer 13

if it is both one-to-one and onto

Question 14

if function is onto

Answer 14

|D| >= |T|

Question 15

if function is one-to-one

Answer 15

|D| <= |T|

Question 16

if function is bijection

Answer 16

|D| <= |T| and |D| >= |T|

Question 17

inverse

Answer 17

f is obtained by exchanging the first and second entries

f = (x,y), f^-1 = (y,x), if bijection

Question 18

when does a function have an inverse?

Answer 18

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

Question 19

composition

Answer 19

the process of applying a function to the result of another function

Question 20

identity function

Answer 20

always maps a set onto itself and maps every element onto itself eg. IvA: A → A, Iva(a) = a for all a ∈ A