6. Module No # 05
This chapter discusses countably infinite sets and transitions into uncountably infinite sets, focusing on Cantor’s diagonalization argument. The proof shows that the set of all binary strings of infinite length is uncountable by demonstrating that for any proposed enumeration, there will always be at least one string that is omitted. Different infinite sets such as {0, 1} and the real numbers between 0 and 1 are explored in further detail, reinforcing the concept of uncountability.
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Sections
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What we have learnt
- Countably infinite sets have the same cardinality as the set of positive integers.
- Cantor’s diagonalization argument is a method to prove the existence of uncountable sets.
- The set of real numbers is uncountable due to its inclusion of irrational numbers, which cannot be enumerated.
Key Concepts
- -- Countably Infinite Set
- A set whose elements can be put into a one-to-one correspondence with the positive integers.
- -- Uncountably Infinite Set
- A set that cannot be placed in one-to-one correspondence with the set of positive integers, implying there are more elements than can be enumerated.
- -- Cantor's Diagonalization Argument
- A proof technique used to demonstrate the existence of uncountable sets by showing that any list of the elements will miss at least one element.
- -- Bijection
- A one-to-one correspondence between two sets, indicating they have the same cardinality.
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