Index 1. SETS 3.2. COUNTABLE AND UNCOUNTABLE SETS 3.2.1. Definition. A set is countably infinite ... more Index 1. SETS 3.2. COUNTABLE AND UNCOUNTABLE SETS 3.2.1. Definition. A set is countably infinite (or denumerable) if it is cardinally equivalent to the set N of natural numbers. A bijection from N onto a countably infinite set S is an enumeration of the elements of S. A set is countable if it is either finite or countably infinite. If a set is not countable it is uncountable. 3.2.2. Example. The set E of even integers in N is countable. The first proposition of this section establishes the fact that the "smallest" infinite sets are the countable ones. 3.2.3. Proposition. Every infinite set contains a countably infinite subset. If we are given a set S which we believe to be countable, it may be extremely difficult to prove this by exhibiting an explicit bijection between N and S. Thus it is of great value to know that certain constructions performed with countable sets result in countable sets. The next five propositions provide us with ways of generating new countable sets from old ones. In particular, we show that each of the following is countable. (1) Any subset of a countable set. (2) The range of a surjection with countable domain. (3) The domain of an injection with countable codomain. (4) The product of any finite collection of countable sets. (5) The union of a countable family of countable sets. 3.2.4. Proposition. If S ⊆ T where T is countable, then S is countable. The preceding has an obvious corollary: If S ⊆ T and S is uncountable, then so is T. 3.2.5. Proposition. If f : S → T is injective and T is countable, then S is countable. 3.2.6. Proposition. If f : S → T is surjective and S is countable, then T is countable. 3.2.7. Example. The set N × N is countable. 3.2.8. Example. The set {x ∈ Q : x > 0} is countable. 3.2.9. Proposition. If S and T are countable sets, then so is S × T. 3.2.10. Corollary. If S 1 ,. .. , S n are countable sets, then S 1 × • • • × S n is countable. Finally we show that a countable union of countable sets is countable. 3.2.11. Proposition. Suppose that A is a countable family of sets and that each member of A is itself countable. Then A is countable. 3.2.12. Example. The set Q of rational numbers is countable. By virtue of 3.2.4-3.2.11 we have a plentiful supply of countable sets. We now look at an important example of a set which is not countable. 3.2.13. Example. The set R of real numbers is uncountable. Proof. We take it to be known that if we exclude decimal expansions which end in an infinite string of 9's, then every real number has a unique decimal expansion. By (the corollary to) proposition 3.2.4 it will suffice to show that the open unit interval (0, 1) is uncountable. Argue by contradiction: assume that (0, 1) is countably infinite. (We know, of course, from example 3.1.10 that it is not finite.) Let r 1 , r 2 , r 3 ,. .. be an enumeration of (0, 1). For each j ∈ N the number r j has a unique decimal expansion 0.r j1 r j2 r j3. .. . Construct another number x = 0.x 1 x 2 x 3. .. as follows. For each k choose x k = 1 if r kk = 1 and x k = 2 if r kk = 1. Then x is a real number between 0 and 1, and it cannot be any of the numbers r k in our enumeration (since it differs from r k at the k th decimal place). But this contradicts the assertion that r 1 , r 2 , r 3 ,. .. is an enumeration of (0, 1).
Index 1. SETS 3.2. COUNTABLE AND UNCOUNTABLE SETS 3.2.1. Definition. A set is countably infinite ... more Index 1. SETS 3.2. COUNTABLE AND UNCOUNTABLE SETS 3.2.1. Definition. A set is countably infinite (or denumerable) if it is cardinally equivalent to the set N of natural numbers. A bijection from N onto a countably infinite set S is an enumeration of the elements of S. A set is countable if it is either finite or countably infinite. If a set is not countable it is uncountable. 3.2.2. Example. The set E of even integers in N is countable. The first proposition of this section establishes the fact that the "smallest" infinite sets are the countable ones. 3.2.3. Proposition. Every infinite set contains a countably infinite subset. If we are given a set S which we believe to be countable, it may be extremely difficult to prove this by exhibiting an explicit bijection between N and S. Thus it is of great value to know that certain constructions performed with countable sets result in countable sets. The next five propositions provide us with ways of generating new countable sets from old ones. In particular, we show that each of the following is countable. (1) Any subset of a countable set. (2) The range of a surjection with countable domain. (3) The domain of an injection with countable codomain. (4) The product of any finite collection of countable sets. (5) The union of a countable family of countable sets. 3.2.4. Proposition. If S ⊆ T where T is countable, then S is countable. The preceding has an obvious corollary: If S ⊆ T and S is uncountable, then so is T. 3.2.5. Proposition. If f : S → T is injective and T is countable, then S is countable. 3.2.6. Proposition. If f : S → T is surjective and S is countable, then T is countable. 3.2.7. Example. The set N × N is countable. 3.2.8. Example. The set {x ∈ Q : x > 0} is countable. 3.2.9. Proposition. If S and T are countable sets, then so is S × T. 3.2.10. Corollary. If S 1 ,. .. , S n are countable sets, then S 1 × • • • × S n is countable. Finally we show that a countable union of countable sets is countable. 3.2.11. Proposition. Suppose that A is a countable family of sets and that each member of A is itself countable. Then A is countable. 3.2.12. Example. The set Q of rational numbers is countable. By virtue of 3.2.4-3.2.11 we have a plentiful supply of countable sets. We now look at an important example of a set which is not countable. 3.2.13. Example. The set R of real numbers is uncountable. Proof. We take it to be known that if we exclude decimal expansions which end in an infinite string of 9's, then every real number has a unique decimal expansion. By (the corollary to) proposition 3.2.4 it will suffice to show that the open unit interval (0, 1) is uncountable. Argue by contradiction: assume that (0, 1) is countably infinite. (We know, of course, from example 3.1.10 that it is not finite.) Let r 1 , r 2 , r 3 ,. .. be an enumeration of (0, 1). For each j ∈ N the number r j has a unique decimal expansion 0.r j1 r j2 r j3. .. . Construct another number x = 0.x 1 x 2 x 3. .. as follows. For each k choose x k = 1 if r kk = 1 and x k = 2 if r kk = 1. Then x is a real number between 0 and 1, and it cannot be any of the numbers r k in our enumeration (since it differs from r k at the k th decimal place). But this contradicts the assertion that r 1 , r 2 , r 3 ,. .. is an enumeration of (0, 1).
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