Page 328 - HOW TO PROVE IT: A Structured Approach, Second Edition

P1: PIG/KNL P2: IWV/
0521861241c07 CB996/Velleman October 20, 2005 1:12 0 521 86124 1 Char Count= 0

314 Infinite Sets
17. Suppose A is denumerable and R is a partial order on A. Prove that R
∗
can be extended to a total order on A. In other words, prove that there
is a total order T on A such that R ⊆ T . Note that we proved a similar
theorem for finite A in Example 6.2.2. (Hint: Since A is denumerable,
we can write the elements of A in a list: A ={a 1 , a 2 , a 3 ,...}.Now,us-
ing exercise 2 of Section 6.2, recursively define partial orders R n , for
+
n ∈ N, so that R = R 0 ⊆ R 1 ⊆ R 2 ⊆ ... and ∀i ∈ I n ∀ j ∈ Z ((a i , a j ) ∈
R n ∨ (a j , a i ) ∈ R n ). Let T =∪ n∈N R n .)
18. Suppose A is finite and B ⊆ A. By exercise 8, B and A \ B are both finite.
Prove that |A \ B|=|A|−|B|. (In particular, if a ∈ A then |A \{a}| =
|A|− 1. We used this fact in several proofs in Chapter 6; for example, we
used it in Examples 6.2.1 and 6.2.2.)
19. Suppose n is a positive integer and for each i ∈ I n , A i is a finite set. Also,

assume that ∀i ∈ I n ∀ j ∈ I n (i = j → A i ∩ A j = ∅). Prove that ∪ i∈I n A i
n
A i |= |A i |.
i=1
is finite and |∪ i∈I n
∗
20. (a) Prove that if A and B are finite sets, then A × B is finite and |A ×
B|=|A|·|B|. (Hint: Use induction on |B|. In other words, prove
the following statement by induction: ∀n ∈ N∀A∀B(if A and B are
finite and |B|= n, then A × B is finite and |A × B|=|A|· n). You
may find Theorem 4.1.3 useful.)
(b) A meal at Alice’s Restaurant consists of an entree and a dessert. The
entree can be either steak, chicken, pork chops, shrimp, or spaghetti,
and dessert can be either ice cream, cake, or pie. How many different
meals can you order at Alice’s Restaurant?
21. For any sets A and B, the set of all functions from A to B is de-
A
noted B.
A
B
(a) Prove that if A ∼ B and C ∼ D then C ∼ D.
(b) Prove that if A, B, and C are sets and A ∩ B = ∅, then A∪B C ∼
B
A C × C.
A A
(c) Prove that if A and B are finite sets, then B is finite and | B|=
|B| . (Hint: Use induction on |A|.)
|A|
(d) A professor has 20 students in his class, and he has to assign a grade
of either A, B, C, D, or F to each student. In how many ways can the
grades be assigned?
22. Suppose |A|= n, and let F ={ f | f is a one-to-one, onto function from
∗
I n to A}.
(a) Prove that F is finite, and |F|= n!. (Hint: Use induction on n.)
(b) Let L ={R | R is a total order on A}. Prove that F ∼ L, and therefore
|L|= n!.
(c) Five people are to sit in a row of five seats. In how many ways can
they be seated?