Homework 6: Functions, Summations, but only some Cardinality

Due 05.Mar.01 (Tue)

Before you tackle the homework, remind yourself of our general hw policies. In particular, you don't need to show your work when only a short answer is required (though doing so might garner more partial credit if the answer is wrong).

Reading: Rosen 1.8 (functions), 3.2 (summations and cardinality).

1. (2pts) Section 1.8, #2. (Is f a function?)
   [Don't confuse Section 1.8 with the other end-of-chapter Review Questions!]
   Note: the square-root symbol denotes a (single-valued) function, the positive number which squares appropriately. (To think about: in the complex plane, how can you specify which of the possible answers is returned by sqrt? I.e. why is sqrt(-1) = i, not -i?)
2. (2pts) Section 1.8, #8 (computing w/ floor, ceiling).
3. (2pts) Section 1.8, #14 (f onto?).
4. (2pts) Section 1.8, #16 (Exhibit types of functions).
   As usual, give the simplest answers you can come up with.
5. (2pts) Section 1.8, #18 (f bijection?).
6. (3pts) Section 1.8, #48 (|[a,b] ∩ Z|).
7. (4pts) Rosen 3.2, #16 (Summations)
   See Rosen 3.2 and/or some summation hints (ps, pdf).
8. (4pts) Rosen 3.2, #18 (Double summations), but replace the indices 2,3 with 200,300 respectively. (Leave exponents unchanged.)
   See Rosen 3.2 and/or some summation hints (ps, pdf).
9. (4pts) Rosen 3.2, #26 (Sum of cube roots).
   You may assume that m is a perfect cube. Hints:
   • As always, work out some examples by hand, first. When you group like terms, how many do you group each time?
   • Don't get too hung up on Rosen's hint? (It wasn't helpful to me.)
10. (1pt extra credit) Extend the previous problem to when m is not necessarily a perfect cube.
11. Deferred to a later hw: (6pts) Rosen 3.2, #32 (Specific sets countable?).
    Write your function in standard mathematical notation, not English. (Hint: Standard notation includes floor and ceiling, as well as a case statement (big-curly-left-bracket).)
    Be sure to convince yourself that the correspondence you give really is both one-to-one and onto!
12. (2pts extra-credit) subsets of countable sets
    Prove that if A is an infinite set, and f:N → A is onto, then A is countable.
    Construct your bijection explicitly (unlike Rosen's answer to his Exercise #35, which informally describes a construction).
13. (3pts extra-credit required) Rosen 3.2, #36 (Union of countable sets).
    Hint: if you don't appeal to the previous extra-credit problem, you have probably overlooked a detail!