Page 327 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Equinumerous Sets 313
7. Suppose A and B are sets and A is finite. Prove that A ∼ B iff B is also
finite and |A|=|B|.
∗ 8. (a) Prove that if n ∈ N and A ⊆ I n , then A is finite and |A|≤ n. Further-
more, if A = I n , then |A| < n.
(b) Provethatif A isfiniteand B ⊆ A,then B isalsofinite,and|B|≤|A|.
Furthermore, if B = A, then |B| < |A|.
9. Suppose B ⊆ A, B = A, and B ∼ A. Prove that A is infinite.
10. Prove that if n ∈ N, f : I n → B, and f is onto, then B is finite and
|B|≤ n.
11. Suppose A and B are finite sets and f : A → B.
(a) Prove that if |A| < |B| then f is not onto.
(b) Prove that if |A| > |B| then f is not one-to-one. (This is sometimes
called the Pigeonhole Principle, because it means that if n pigeons
are put into m pigeonholes, where n > m, then some pigeonhole must
contain more than one pigeon.)
(c) Prove that if |A|=|B| then f is one-to-one iff f is onto.
12. Show that the function f : Z × Z → Z defined by the formula
∗ + + +
(i + j − 2)(i + j − 1)
f (i, j) = + i
2
is one-to-one and onto.
13. Complete the proof of part 2 of Theorem 7.1.2 by showing that if f : A →
B and g : C → D are one-to-one, onto functions, A and C are disjoint,
and B and D are disjoint, then f ∪ g is a one-to-one, onto function from
A ∪ C to B ∪ D.
∗
14. In this exercise you will complete the proof of 3 → 1 of Theorem 7.1.5.
Suppose B ⊆ Z and B is infinite. We now define a function f : Z → B
+
+
by recursion as follows:
+
For all n ∈ Z ,
f (n) = the smallest element of B \{ f (m) | m ∈ Z , m < n}.
+
Of course, the definition is recursive because the specification of f (n)
refers to f (m) for all m < n.
+
(a) Supposen ∈ Z .Thedefinitionof f (n)onlymakessenseifwecanbe
sure that B\{ f (m) | m ∈ Z , m < n} = ∅, in which case the well-
+
ordering principle guarantees that it has a smallest element. Prove
+
that B\{ f (m) | m ∈ Z , m < n} = ∅. (Hint: See exercise 10.)
(b) Prove that for all n ∈ Z , f (n) ≥ n.
+
(c) Prove that f is one-to-one and onto.
15. Prove that if B ⊆ A and A is countable, then B is countable.
16. Prove that if B ⊆ A, A is infinite, and B is finite, then A \ B is infinite.