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Cardinality

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1. Find the cardinality of the set of all irrational numbers, and prove your answer is correct.

2a. Is there a line in the x-y plane such that both coordinates of every point on the line are rational? Prove your answer is correct.

2b. Find the cardinality of the set of all complex numbers, and justify your answer.

3a. What is the cardinality of the set of all finite sets subsets of R? Prove your answer is correct.

3b. Find the cardinality of the set of all points in the plane which have one rational coordinate and one irrational coordinate, and justify your answer.

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The expert examines cardinality for irrational numbers.

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1,Find the cardinality of the set of all irrational numbers, and prove your answer is correct.
Solution: The cardinality of set of irrational numbers is infinity. The set of irrational numbers is defined as:
.

Since there are infinite such x which cannot be written in terms of p and q, the cardinality becomes infinite.
New Notation: The set of irrational numbers has cardinality = C ("the infinity of the continuum").

2a, Is there a line in the x-y plane such that both coordinates of ...

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