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I will try to update this page every lecture, to announce the due dates for the assignments from each section covered. The problems listed below are tentative (and may change!!) until these due dates have been announced. Always recheck that you have done the problems actually assigned, before submitting them!
Assignments
(Numbers refer to 4th Edition. If you have an earlier edition, please compare with the 4th edition to see what problems you need to do.)
EXTRA CREDIT 1: (Due Mon, Sep 9)
What was the mistake in the inductive “proof” done in class to show any n objects were the same? Be very specific.
HW1 (Due Wed, Sep 11)
Problem A0: Let A1={3k: k ∈ N}, A2={3k-1: k ∈ N}, A3={3k-2: k ∈ N}. Also, let E be the set of all even natural numbers.
Give the intersection of E with each of A1, A2, and A3. State each as a set of the form S={formula involving k, k ∈ N}.
What is the union of A1, A2 and A3?
Sec 1.2: #1,5,7,11,14
Review answers to “Logic” problems (see BBd under “Course Documents/Class Handouts and Solutions”). No submission needed.
HW2 (Due Wed, Sep 18)
Sec 2.1: # 7,8(b),9
Sec 2.2: # 5, 6(a), 16, 17
Problem A1: First show that if x is in V_ε(a) and y is in V_ε(b), then x+y has to be in V_(2ε)(a+b). Next, by counterexample, show that x+y does not have to be in V_ε(a+b). (Give actual values of a, b, x, y and ε for this counterexample)
HW 3 (Due Wed, Sep 25)
Sec 2.3: # 1, 4(Justify!), 8, 9, 11
(NOTE: Use only arguments from this section, not Sec 2.4)
Sec 2.4: # 2, 3, 5, 19
HW 4 (Due Wed, Oct 2)
NOTE: Please make a copy of your submission for yourself. This will help you study for the test on Oct 7. The solutions will be available to you on Bbd after 1 pm on Thursday, Oct 3.
HW assigned in class: Give an algebraic formula for a bijection between the set of integers and the set of naturals.
Sec 1.3: # 4, 12, 13 (make sure you check hint at back)
Also, #1 through 6 from: Cardinality Problems.
EXTRA CREDIT: Due Date Mon, Oct 14
You are given an unbiased coin for which the probability of getting a head (or a tail) in any toss is exactly 1/2. You toss it repeatedly, to generate an infinite sequence (x_n) whose terms are given by
x_n=(number of heads in n tosses)/n
(i.e. x_1=(number of heads in 1st toss)/1, x_2=(number of heads in first 2 tosses)/2, etc).
Say whether or not Lim x_n = 1/2 in the sense of definition 3.1.3, justifying your answer.
HW 5 (Due Wed, Oct 16)
Sec 3.1 #5(b),(d), 7, 18 (NOTE: Only use the definition in problems from this section, not theorems from Sec 3.2)
Sec 3.2 # 4,7,9,15, 22
EXTRA CREDIT: Due Date Mon, Oct 21
Suppose (x_n) is a convergent sequence with limit L>0. Use the definition of convergence to show that log(x_n) converges to log(L). (For simplicity, you can assume that x_n > L for each n, though this is not, strictly speaking, necessary.) Hint: Use the mean value theorem from Calculus.
HW 6 (Due Wed, Oct 23)
Sec 3.2 # 5 (Use only results from Sec 3.1 and 3.2, not 3.3 or 3.4)
Sec 3.3 # 3,5,7,9, 12(c)(d)
HW 7 (Due Wed, Oct 30)
Sec 3.4: # 3,4,9,12 (Also, try #14, but do not submit.)
Sec 3.5 # 2(a), 3(b), 4, 5, 9
EXTRA CREDIT: Due Date Mon, Nov 4
Given a series S=Sum(a_n), one can define a subseries S’=Sum(a_n_k). Prove or Disprove: Every positive p-series, p>0, always has a convergent subseries.
HW 8 (Due Wed, Nov 6)
Sec 3.5 #12, 13
Sec 3.7 # 3(b), 4,5, 10, 11, 12,13, 14
EXTRA CREDIT (Due Wed, Nov 13)
King Lear Problem: We know that any angle can be bisected using straight edge and compass. (a) How would you apply this bisection technique repeatedly to approximately trisect the angle? (b) By considering the limit of an infinite series, prove that you converge to the trisected angle as the number of repetitions approaches infinity.
HW 9 (Due Wed, Nov 20)
Sec 4.1 #6,7 (use definition, not sequential criterion),10(a),12(d), 14
Sec 4.2 # 4,5,9, 11(d)
HW 10 (Due Mon, Dec 2, but try to submit on Wed, Nov 27)
Sec 5.1 # 3, 7,11,12,13
Sec 5.2 #3,5
Sec 5.3 # 4,6
HW 11 (Due Wed, Dec 4)
Sec 6.1 # 2,4,9,10,13
Sec 6.2 #6,8,9,17
EXTRA CREDIT (Due Mon, Dec 9)
Sec 11.1 # 5,6,8
Problem L1: True or false (justify in each case):
(1) If A is closed, then removing a single point from A will always give a set that is no longer closed.
(2) If A is open, then removing a single point from A will always give a set that is no longer open.
Problem L2: Show that if the set A is closed, then so is the set B={2x, x in A}. (Hint: Use 11.1.7)