# Cardinality, countability

2-12 through 2-14.

Exercise 2-12. Consider the integers in the arrangement 0, 1, -1, 2, -2, 3, -3, ... Let n E N. Which integer occupies the 2n position? The 2n+ 1 position? Prove that I and N can be put into one-to-one correspondence.

From Exercise 2-12, I has N0 integers. As we have seen, adding one new members to a countable infinity does not change the cardinal number of a set.

Exercise 2-12. Prove that adding a finite number n of new members to a countably infinite set does not change its cardinal numbers. (Find a one-to-one correspondence as in the hotel illustration proceeding Definition 2-10.)

2-12, Not sure what they are asking with position of 2n, 2n+1 position of the integers. The third part, proof on how to get the integers and N into one to one, can I just show ordering hence; 1,2,3,4, ... and 0,1,-1,2,-2,... or should there be more

Exercise 2-14. Prove that n countably infinite sets can be put into one-to-one correspondence with one countably infinite set. (Use the technique of Exercise 2-12.)

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#### Solution Preview

See the attachment.

Ex. 2-12. The number n occupies the 2n position, and the number (-n) occupies the (2n+1) position.

The given arrangement gives a one-to-one correspondence. Indeed, different integers are located into different positions. Moreover, any position contains some integer.

Ex.2-13. Let identify the elements of a countably infinite set with ...

#### Solution Summary

This solution answers various questions regarding cardinality and countability.