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Magazine Problem 9
Prove or disprove: The set \(\\{0,1\\} \times \mathbb{N}\) is countably infinite.
Problem 9
Prove that if \(A\) and \(B\) are finite sets with \(|A|=|B|,\) then any injection \(f: A \rightarrow B\) is also a surjection. Show this is not necessarily true if \(A\) and \(B\) are not finite.
Problem 10
Prove or disprove: The set \(A=\left\\{\frac{\sqrt{2}}{n}: n \in \mathbb{N}\right\\}\) countably infinite.
Problem 10
Show that the two given sets have equal cardinality by describing a bijection from one to the other. Describe your bijection with a formula (not as a table). \(\\{0,1\\} \times \mathbb{N}\) and \(\mathbb{Z}\)
Problem 10
Prove that if \(A\) and \(B\) are finite sets with \(|A|=|B|\), then any surjection \(f: A \rightarrow B\) is also an injection. Show this is not necessarily true if \(A\) and \(B\) are not finite.
Problem 11
Describe a partition of \(\mathbb{N}\) that divides \(\mathbb{N}\) into eight countably infinite subsets.
Problem 12
Describe a partition of \(\mathbb{N}\) that divides \(\mathbb{N}\) into \(\aleph_{0}\) countably infinite subsets.
