Question

3. Give an example of a function
(a) f : Z → N that is both 1-1 and onto.
(b) f : N → Z that is both 1-1 and onto

Isn't impossible to do both? I mean if you draw a venn diagram, one can see that one set is bigger than the other ( therefore not allowing it to be "onto") and at the same time the inverse wouldn't allow a one - to - one relationship. Any help on what I'm doing wrong?

Answer 1

does your text define 0 as a natural number?

Answer 2

Z and N are both countably infinite, so they have the same number of elements

n(0,1, 2, 3, 4, 5, 6,7, 8,...)
z(0,1,-1,2,-2,3,-3,4,-4,...)

maps n to z

Answer 3

i thought n was from 1--> infinity and
z was from -infinity --> infinity excluding decimals

Answer 4

i spose it depends on how you 'define' the function

this paranthetical top/bottom is a functional definition

Answer 5

even so, if n starts at 1, we can still develop an onto relationship

Answer 6

can we define it in terms of some equation? im not sure that we can. but a function need not be an equation

Answer 7

yeah but it can't be one to one. If Z is from 0 n 0

Answer 8

your mistaking elements, with cardinality

Answer 9

mhmm what do you mean? I thought that it represented all numbers within one set. if the set of integers is greater, than it will never be one - to - one as there are too many of one to fill the other, no?

Answer 10

f:A to B, is one to one and onto

Answer 11

why?