Notesale: Turn your study into money

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complement of a countable set is also uncountable
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The complement of a group is defined as the
set of all features that are not in that set
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Since the set of real numbers is uncountable, and the complement of a countable set is a subset
of the group of real numbers, it follows that the addition of a countable set is also uncountable
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And since the complement of a
countable set is a subset of the uncountable set of real numbers, it must also be uncountable
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To prove that the Cartesian product of two countable sets is countable, we can use the fact
that a group is countable if a one-to-one correspondence exists between its elements and the
natural numbers
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In that case, we can create a
one-to-one correspondence between the details of their Cartesian product A x B and the
natural numbers by listing all the ordered pairs (a, b) where a is an element of A and b is an
element of B in lexicographic order (i
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, (a1, b1), (a1, b2),
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, (an,
bn) )
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The Cartesian product of two sets, A and B, denoted by A x B, is the set of all ordered pairs (a,b)
where a is an element of A and b is an element of B
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This means that each group component can be paired with a unique natural
number, and every natural number corresponds to a component of the set
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Lexicographic order is a way of arranging words or other strings in a dictionary or a phonebook,
where each piece is ordered first by its first letter, then by its second letter, and so on
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So, given that we have two countable sets, A and B, by listing all the ordered pairs (a, b) of the
Cartesian product A x B in lexicographic order, we can pair each element of A x B with a unique
natural number, which is the definition of countable
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To prove that the set of all real numbers is uncountable, one way is to use the diagonal
argument
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We can do this
by creating a table of all real numbers in the form of decimal expansions
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This number is different from every number in the list and thus cannot be on the list, which
shows our assumption is incorrect
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The proof that the set of all real numbers is uncountable is based on the idea of contradiction
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To prove this assumption is false, a method is used to construct an actual number that cannot
b

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