Question

4a. Sketch a proof that the Cartesian product ? × ? is countably infinite. 4b. Write down a set which has the same cardinality as ?(? × ?). 4c. Prove that for each n ? 1, the set ?(n) of all subsets of ? containing exactly n elements, is countable. [Hint: Use induction together with the fact that the countable union of countable sets is countable.]

          4a. Sketch a proof that the Cartesian product ? × ? is countably infinite.
4b. Write down a set which has the same cardinality as ?(? × ?).
4c. Prove that for each n ? 1, the set ?(n) of all subsets of ? containing
exactly n elements, is countable. [Hint: Use induction together with the
fact that the countable union of countable sets is countable.]

4a. Sketch a proof that the Cartesian product ? × ? is countably infinite.
4b. Write down a set which has the same cardinality as ?(? × ?).
4c. Prove that for each n ? 1, the set ?(n) of all subsets of ? containing
exactly n elements, is countable. [Hint: Use induction together with the
fact that the countable union of countable sets is countable.]

Added by Brenda V.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

07/07/2023

Step 1

To show that the Cartesian product $N \times N$ is countably infinite, we need to find a bijection between $N \times N$ and $N$. We can use the Cantor's pairing function, which is defined as: $$\pi(x, y) = \frac{(x + y)(x + y + 1)}{2} + y$$ This function maps Show more…

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4a. Sketch a proof that the Cartesian product N x N is countably infinite. 4b. Write down a set which has the same cardinality as P(N x N). 4c. Prove that for each n ≥ 1, the set A(n) of all subsets of N containing exactly n elements, is countable. [Hint: Use induction together with the fact that the countable union of countable sets is countable ]