Julian Homework 3

Bijections

1a. Everything correct except don't write | | = | | = infinity. Dude...no need to write same size when you already proved there is a surjection+injection = bijection. The problem is the symbol infinity...there are different infinities! For example, the countable infinity, aleph 0 and an uncountable infinity aleph 0. The natural numbers and real numbers are both infinite sets (with respective infinities aleph 0 and aleph 1), but no bijection exist! To learn more, see

http://en.wikipedia.org/wiki/Aleph_number

1b. See 1a.

2. Duh.

3a, b. Good! Explicit definitions = awesomeness.

4. Bah! Wanted FTA, but this works too.

5. See 1a.

6. I gotta give harder questions.

7.1. Nice!

7.3. Hah, used same trick as in 3a. Love the math vocab/jargon. "Induces". Cool. We can do the labelling trick with more things than a chessboard in order to relate to a permutation group. Maybe I should teach Cayley's theorem...

7.5. Good catch with the inverse ordering. ABSOLUTELY GREAT: inverses are 1:1, surjective, and unique! GREAT JOB!

7.8 Good isomorphism check, Nice job!