# Help with a proof please!

• Apr 25th 2008, 06:34 PM
calcprincess88
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?

• Apr 25th 2008, 08:46 PM
TheEmptySet
Quote:

Originally Posted by calcprincess88
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?

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.
• Apr 25th 2008, 09:59 PM
Isomorphism
Quote:

If a set is equivelent to any of its proper subsets, then the set is infinite
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 Math
• Apr 27th 2008, 11:07 AM
calcprincess88
Thanks for your help! I'll try to do it and see what I can come up with and if I need anymore help I'll come back! Thanks again!