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The problems listed below are tentative (and may change!!). 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.)
C1: (Due Wed, Feb 3)
What was the mistake in the inductive “proof” that shows any n numbers are the same? Be very specific.
HW1 (Due Mon, Feb 8)
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
HW2 (Due Mon, Feb 15)
Sec 2.1: # 7,8(b),9
Sec 2.2: # 5, 6(a), 16, 17
HW3 (Due Mon, Feb 22)
Sec 2.3: # 1, 4(Justify!), 8, 9, 11
(NOTE: Use only arguments from this section, not Sec 2.4)
Sec 2.4: # 2 (Justify!), 3, 5, 19
C2 (Due Wed, Feb 24)
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.)
HW4 (Due Mon, Mar 1)
Sec 1.3: # 4, 12, 13 (make sure you check hint at back)
Also, #1 through 5 from: Cardinality Problems
C3 (Due Wed, Mar 10)
You are given an unbiased coin for which the probability of getting a head H (or a tail T) in any toss is exactly 1/2. You toss it repeatedly, to generate an infinite sequence of heads and tails, for instance HTHHTH….
(a) Let S be the set of all such sequences. What is the cardinality of S? Justify.
(b) Suppose, for any sequence in S, you generate a numerical sequence (xn) whose terms are given by xn =(number of heads in first n tosses)/n
(i.e. x1 =(number of heads in 1st toss)/1, x2 =(number of heads in first 2 tosses)/2, x3 =(number of heads in first 3 tosses)/3, etc).
What sequence (xn) would be generated by THTHTHTHTH…. (i.e if you kept getting a tail followed by a head)?
(c) Would (xn) from part (b) converge to 1/2? Prove your answer.
(d) Will every possible sequence of heads and tails in S give rise to an (xn) that converges to 1/2? Justify your answer.
HW 5 (Due Mon, Mar 22)
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
HW6 (Due Mon, Mar 29)
Sec 3.3 # 3,5,7,9, 12(c)(d)
HW 7 (Due Mon, Apr 5)
Sec 3.4: # 3,4,9,12
Sec 3.5 # 2(a), 3(b), 4, 5, 9
C4 (Due Wed, Apr 7)
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 8 (Due Mon, Apr 12)
Sec 3.5 #12, 13
Sec 3.7 # 3(b), 4,5, 10, 11, 12,13, 14
C5: (Due Wed, Apr 21)
Given a series S=Σan, one can define a subseries S’=Σan_k for any subsequence (an_k) of (an). Prove or Disprove: Every positive p-series, p>0, always has a convergent subseries.
HW 9 (Due Mon, Apr 26)
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, May 3)
Sec 5.1 # 3, 7,11,12,13
Sec 5.2 #3,5
Sec 5.3 # 4,6
HW 11 (Due Mon, May 10)
Sec 6.1 # 2,4,9,10,13
Sec 6.2 #6,8,9,17
C6: (Due Wed, May 12)
(Will be based on Sec 11.1)