Saturday, 13 August 2016

elementary set theory - How to prove a set is countably infinite



The problem states that A is countably infinite and element b is not in A. It then asks to show that A union {b} is countable infinite.



I'm pretty sure I need to find a bijection between the union and the set of all positive natural numbers, I'm just having trouble figuring out where to go after introducing said function, or how to prove such a function is one to one and onto. Any pointers?


Answer



Let a1,a2,a3, be a sequence containing all members of the set A.



Then b,a1,a2,a3, is a sequence containing all members of the set A{b}.

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