To do these problems you should understand the basic ideas of cardinality via bijections. Also, you need to read and understand the proof of Theorem 2.5.5, which was described in the talk.
(Note that the talk is on YouTube if needed:
http://www.youtube.com/watch?v=kBS_cNHvnBE)
1. Show that the sets S={1,3,5,7…} and T={1^2, 2^2, 3^2,…} have the same cardinality by exhibiting a bijection between them. You don’t need to formally prove that it is a bijection, but you do need to give a formula f:S→T.
2. Show that the intervals [-3,1] and [4,6] have the same number of points, by exhibiting a bijection between them. Again, no proof needed that it is a bijection, but you do need to give a formula g:[-3,1]→[4,6].
3. To show that the set of all real numbers between 0 and 1 had a higher cardinality than N_0, we assumed first that they had the *same* cardinality. Then we showed that there was at least one number which was not counted in the list.
Suppose now that we *only* consider those numbers between 0 and 1 which have a decimal expansion consisting of 0’s and 1’s. (e.g. 0.10101…, 0.0110010.., etc.) Show that this set *also* has a cardinality greater than N_0 (once again by the same kind of construction).
4. Where does the proof that showed all reals between 0 and 1 are uncountable break down when applied to all rationals between 0 and 1? (If it didn’t break down, you could show that all rationals between 0 and 1 are also uncountable.)
5. For each set below, say whether it is finite, countably infinite, or uncountable. Justify your answer in each case, giving a brief reason rather than an actual proof.
(a) The points along the circumference of a unit circle.
(b) The carbon atoms in a single page of the textbook.
(c) The different angles that could be formed when two lines intersect (e.g. 30 degrees, 45 degrees, 359.89 degrees, etc….)
(d) All irrationals which are exact square roots of a natural number.
(e) All irrationals of the form a+sqrt(b) where a and b are rational numbers.
(f) The set of all squares that can be drawn within a unit circle.
6. Let a “word” be defined as a finite string of letters from the English alphabet. For example, arrokk, kjhhhsooop, eloootttyg are all “words.” Let S be the set of all such words. Justifying your answer, say whether S is finite, countably infinite or uncountable.