I need help with this question. It says to use Corollary 5.10(meaning this: A finite set is not equivalent to any of its proper subsets) to show that the rationals Q are infinite.
How would I go about doing this?
Thanks for your help!
I need help with this question. It says to use Corollary 5.10(meaning this: A finite set is not equivalent to any of its proper subsets) to show that the rationals Q are infinite.
How would I go about doing this?
Thanks for your help!
Lets rewrite the Corollary as an if then statement
If a Set is finite then it is not equivelent to any of its proper subsets
$\displaystyle p \implies q$
Use the contrapositive of this corollary.
$\displaystyle \sim q \implies \sim p$
if a set is equivelent to any of its proper subsets, then the set is infinite
See what you can do from here.
Good luck.
Hint: Show that $\displaystyle \mathbb{Z}$ is equivalent to $\displaystyle \mathbb{Q}$. Use the Cantors famous diagonal trick. See Proof that rational numbers are countable - from Homeschool MathIf a set is equivelent to any of its proper subsets, then the set is infinite