Use the following outline to supply proofs for the statementsin Theorem 1.5.8.((i) If A1,A2, . . . Am are each countable sets, then the unionA1 ∪ A2 ∪ · · · ∪ Am is countable.)(a) First, prove statement (i) for two countable sets, A1 and A2. Example1.5.3 (ii) may be a useful reference. Some technicalities can be avoidedby first replacing A2 with the set B2 = A2\A1 = {x ∈ A2 : x /∈ A1}. Thepoint of this is that the union A1 ∪ B2 is equal to A1 ∪ A2 and the setsA1 and B2 are disjoint. (What happens if B2 is finite?)Now, explain how the more general statement in (i) follows.
Use the following outline to supply proofs for the statementsin Theorem 1.5.8.((i) If A1,A2, . . . Am are each countable sets, then the unionA1 ∪ A2 ∪ · · · ∪ Am is countable.)(a) First, prove statement (i) for two countable sets, A1 and A2. Example1.5.3 (ii) may be a useful reference. Some technicalities can be avoidedby first replacing A2 with the set B2 = A2\A1 = {x ∈ A2 : x /∈ A1}. Thepoint of this is that the union A1 ∪ B2 is equal to A1 ∪ A2 and the setsA1 and B2 are disjoint. (What happens if B2 is finite?)Now, explain how the more general statement in (i) follows.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.4: Binary Operations
Problem 9E: 9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of...
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Use the following outline to supply proofs for the statements
in Theorem 1.5.8.((i) If A1,A2, . . . Am are each countable sets, then the union
A1 ∪ A2 ∪ · · · ∪ Am is countable.)
(a) First, prove statement (i) for two countable sets, A1 and A2. Example
1.5.3 (ii) may be a useful reference. Some technicalities can be avoided
by first replacing A2 with the set B2 = A2\A1 = {x ∈ A2 : x /∈ A1}. The
point of this is that the union A1 ∪ B2 is equal to A1 ∪ A2 and the sets
A1 and B2 are disjoint. (What happens if B2 is finite?)
Now, explain how the more general statement in (i) follows.
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