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mathgeek808
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Need help with this proof:
Prove that the set of irrational numbers has the same cardinality as the set of real numbers.
Prove that the set of irrational numbers has the same cardinality as the set of real numbers.
The cardinality of real numbers is the number of elements or points in the set of real numbers, denoted by |R| or c. It represents the size or magnitude of the set.
The cardinality of irrational numbers is also denoted by |R| or c. This means that the number of elements in the set of irrational numbers is equal to the number of elements in the set of real numbers, making their cardinalities equal.
This can be proven using Cantor's diagonal argument, which states that for any infinite set, the cardinality of its power set is greater than the cardinality of the set itself. Since the set of irrational numbers is a subset of the set of real numbers, their cardinalities must be equal.
Proving the equality of these two cardinalities is important in understanding the concept of infinity and sets. It also helps to establish the concept of uncountability, where certain sets have infinitely many elements that cannot be counted.
Yes, there are other sets with equal cardinality to the real numbers, such as the set of algebraic numbers and the set of transcendental numbers. This shows that there are different levels of infinity, all of which have equal cardinalities.