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Flashcards in Functions Deck (20)
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1

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

2

domain (chapter 5)

the set X of f

3

target

the set Y of f

4

f maps x to y can be denoted as...

(x, y) ∈ f OR f(x) = y

5

when is f well-defined

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

6

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 }

7

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

8

floor function

maps real number to nearest integer in the downward direction

9

ceiling function

maps real number to nearest integer in the upward direction

10

nested floor and ceiling functions

perform operation on inside first (like brackets)

11

one-to-one (injective)

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

12

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

13

bijective

if it is both one-to-one and onto

14

if function is onto

|D| >= |T|

15

if function is one-to-one

|D| <= |T|

16

if function is bijection

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

17

inverse

f is obtained by exchanging the first and second entries
f = (x,y), f^-1 = (y,x), if bijection

18

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

19

composition

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

20

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