Due Date: November 20
Please write neatly and leave enough space so I can write comments! Remember that there are solutions to all the odd-numbered exercises in the textbook. I will mainly be looking at your solutions to the evennumbered exercises from the textbook and all the exercises I write, so make sure you attempt most of those.
Please show your work on all problems!
Poker Hands
Exercise 1.
In class, I computed N (pair) and N (2-pair) using combinations. Read the computations of
N (straight flush) and N (flush) in the lecture notes. Compute
(a) N (triple),
(b) N (quadruple),
(c) N (full house),
(d) N (straight) in a similar fashion, by breaking up the procedure of choosing a hand of each type into tasks. Show your work! (Hint: aim for the combinatorial expressions in the table in the lecture notes!)
Section 10.2 (p. 665)
#1, #2, #3 .
Ignore the last sentence of the instructions, which asks you to “identify all isolated and pendant vertices”.
Section 10.3 (p. 676-677)
#34-42 .
If you think a pair of graphs is isomorphic, write down a bijection of the vertices that gives the isomorphism (the bijection of the vertices determines the bijection of the edges). For instance, for #34, a bijection giving an isomorphism is u
1
!→ v
1
, u
2
!→ v
2
, u
3
!→ v
4
, u
4
!→ v
5
, u
5
!→ v
3
.
Can you find the only other bijection of the vertices that gives an isomorphism?
If you think a pair of graphs is not isomorphic, try to give a reason why (the book says to give a rigorous argument, but you don’t have to). For instance, see if the degrees of the vertices match.
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Isomorphism
Exercise 2.
We saw that the right notion of “sameness” for graphs is “isomorphism”, because isomorphism identifies when graphs have the same mathematical structure, namely the same relative arrangement of vertices and edges. Isomorphism is powerful because it allows us to think of graphs as drawings, without worrying about the exact labels of the vertices and edges or exactly how we chose to draw the graph.
In fact, the notion of isomorphism is fundamental in all areas of pure mathematics.
What do you think might be a reasonable definition of “isomorphism” of sets?
(Hint: think about when two sets have the “same” mathematical properties, in particular, the “same” functions. Are the labels of the elements of a set important to the mathematical properties of a set?)
2