Why Read This Book? - Index of

4.7 Infinite Sets 139
The next exercise says that if A and B are disjoint sets, and if there exist
bijections f1 : Nm → A and f2 : Nn → B, then you can construct a bijection
f : Nm+n → A ˙∪ B. Once you have defined such a function, showing it is a bijection
from Nm+n to A ˙∪ B should amount to little more than calling on previous
exercises.
EXERCISE 4.6.7 If A and B are disjoint finite sets with cardinalities m and n,
respectively, then |A ∪ B| = m + n. 8
EXERCISE 4.6.8 If |B| = n and A ⊆ B, then A is finite and satisfies |A| ≤ n. 9
Corollary 4.6.9 Suppose |A| = n and C is any set. Then |A ∩ C| ≤ |A|.
Proof. Since A ∩ C ⊆ A, the result is immediate from Exercise 4.6.8.
EXERCISE 4.6.10 Suppose U is a finite universal set, and A ⊆ U. Then A and
A C are both finite, and |A| + � � A C � � = |U|.
EXERCISE 4.6.11 Suppose |A| = |B| = n, and suppose f : A → B is any oneto-one
function. Then f is onto. 10
EXERCISE 4.6.12 If |A| = m and |B| = n, then |A ∪ B| ≤ m + n. 11
EXERCISE 4.6.13 The union of a finite number of finite sets is finite. 12
4.7 Infinite Sets
The following definition should not be too surprising.
Definition 4.7.1 A set is said to be infinite provided it is not finite.
By negating Definition 4.6.1, we see that a set A is infinite provided that for
every nonnegative integer n, every function from Nn to A fails to be either oneto-one
or onto. In reality, if there were some f : Nn → A that is onto but not
one-to-one, we could create a function f1 : Nm → A for some m