Question

onsider the set of integers ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...} and the set of odd integers \(\mathbb{O}\) = {... -3, -1, 1, 3, ...}. At first glance, one might be tempted into thinking there are half as many odd integers as there are integers. Show that this thinking is erroneous and that there is in fact the same number of odd integers as there are integers by finding a one-to-one and onto function f mapping ℤ into \(\mathbb{O}\) .

Answer 1

How about if you take your function such that \[\begin{align} f: \mathbb{Z}&\longrightarrow \mathbb{Z}' \\ k&\longmapsto 2k+1\end{align}\]

Answer 2

where I'm using \(\mathbb{Z}'\) as the set of odd integers.

Answer 3

waiiiit i messed up somewhere

Answer 4

hmmm idk how to show the O that is like doubled sord of outlined

Answer 5

\(\mathbb{O}\)?

Answer 6

yaaaaa

Answer 7

Type the following: \[\text{(\mathbb{O}\)}\] with an extra "\" in front of the whole thing.

Answer 8

THANNNKKKSSS

Answer 9

but is the answer the same?

Answer 10

It should be. The function I gave you is one-to-one and onto from \(\mathbb{Z}\) to \(\mathbb{O}\). You'll need to actually show this yourself though.

Answer 11

hahah ok let me think abt it