Number positions starting at 1 in the integer list 0,1,-1,2,-2,3,-3,... . At which position does -4 appear?

Let f(n)=2n map N={1,2,...} to E={2,4,...}. Injectivity has been proved. Which fact completes the proof that f is a bijection?

• Every odd number has a positive natural preimage.
• Every e in E has the natural preimage e/2.
• Every integer has a positive natural preimage.
• Every e in E has two different natural preimages.

The rule f(n)=2n pairs positive naturals with positive evens. Where does 7 go?

• 7
• 49
• 14

A proper subset can have a bijection with its whole set even when the whole set is finite.

How many subsets of {a,b,c,d} contain a?

A full hotel has rooms 1,2,3,... . Each current guest moves from room n to room n+1. Which room receives the guest from room 5?

Why does the decimal diagonal proof avoid a constructed decimal ending in an endless tail of 9s?

• A tail of 9s can also be written with a tail of 0s.
• A tail of 9s always represents a number outside (0,1).
• A tail of 9s makes the represented number irrational.
• A tail of 9s means the number has no decimal expansion.

Every infinite subset of N={1,2,3,...} can be listed in increasing order and is countably infinite.

List reals in (0,1) using decimals without a tail of 9s. Construct digit n of d as 2 if digit n of row r_n is 1, and as 1 otherwise. What follows?

• The list includes every real in (0,1)
• The list misses a real in (0,1)
• The list has only rational numbers

Let N={1,2,...} and let f:N -> P(N) be arbitrary. Define D={n in N: n is not in f(n)}. Which argument proves P(N) is uncountable?

• D differs from f(1), so that single comparison rules out every proposed surjection.
• D is a subset of N, so listing the members of N also lists every subset of N.
• For every k, D and f(k) disagree at k, so no such f is onto.
• D is infinite for every f, whereas every output f(k) must be a finite subset of N.