Allen Knutson's Math 113, spring '04, HW #1, due 1/29

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1. How many isomorphisms are there between these two graphs?

2. Find an isomorphism from the first graph to the second graph:

3. Let G,H be two graphs with 5 vertices and 9 edges. Show they are isomorphic. How many isomorphisms are there from G to H?

4. Prove that the number of 1:1 functions from a k element set to a (k+n)-element set is (n+1)(n+2)...(n+k). (Hint: What should you induct on?)

5. Use some symmetries to show that the number of partitions of {a,b,c,d,e} is 52. Please, please do something more systematic than writing out all 52.

6. Prove that a function is an epimorphism iff it's onto. Be as rigorous as you can, and make sure to prove both directions!