10.2 Denumerable Sets
Definition - a set A is called denumerable if |A| = |N|, that is, if A has the same cardinality as the set of natural numbers.
A set is countable if it is either finite or denumerable. Countably infinite sets are then precisely the denumerable sets.
A set that is not countable is called uncountable. An uncountable set is necessarily infinite.
What was difficult for me?
The whole concept of countably infinite was difficult for me. I always thought that being infinite meant that you couldn't count it. But I guess it is. The best example that I understand from it is the sands in the sea or the stars in the sky - there is a limited amount of them. Understanding this concept this way made a lot more sense.
Reading through some of the theorems showed that this section will just be understanding what certain words mean and when to use them. With practice, all will be well.
What did I find interesting?
I love this way of thinking. It's really neat to think like a mathematician - to think of thinks as countable, countably infinity, or uncountable. I also had never heard of the word denumerable before. I'm excited to start using that in my daily vocabulary. I found it interesting to look at the figures they used in the book to see how to classify groups of objects (sets). I'm just fascinated by this way of thinking.
Definition - a set A is called denumerable if |A| = |N|, that is, if A has the same cardinality as the set of natural numbers.
A set is countable if it is either finite or denumerable. Countably infinite sets are then precisely the denumerable sets.
A set that is not countable is called uncountable. An uncountable set is necessarily infinite.
What was difficult for me?
The whole concept of countably infinite was difficult for me. I always thought that being infinite meant that you couldn't count it. But I guess it is. The best example that I understand from it is the sands in the sea or the stars in the sky - there is a limited amount of them. Understanding this concept this way made a lot more sense.
Reading through some of the theorems showed that this section will just be understanding what certain words mean and when to use them. With practice, all will be well.
What did I find interesting?
I love this way of thinking. It's really neat to think like a mathematician - to think of thinks as countable, countably infinity, or uncountable. I also had never heard of the word denumerable before. I'm excited to start using that in my daily vocabulary. I found it interesting to look at the figures they used in the book to see how to classify groups of objects (sets). I'm just fascinated by this way of thinking.
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