Quizzes for Proofs from Book

• Chapter 5 Quiz.
  1. Define: ring, division ring, field, phi_d(x), the class formula, centralizer
  2. Give examples of groups or rings whose center is: - trivial - the whole group - somewhere in between
  3. Describe the nth roots of unity as a group
  4. Compute in H (the quaternions): (i + j + k + 1) * (j + k)
• Chapter 6 Quiz. For each of the following statements indicate whether it is TRUE or FALSE.
  1. e^z is irrational for every z in |R.
  2. e^z is irrational for every z in |Q.
  3. Sum n ≥ 1 (n(n+1))^-1 is irrational.
  4. Sum n ≥ 0 (n+1)^-2 is irrational.
  5. Sum k ≥ 0 (2k+1)^-2 - (3/4) Sum k ≥ 1 k^-2 is irrational.
  6. lim (1 + 1/n)^(2n) is irrational as n -> infinity.
  7. Sum k ≥ 0 (2k+2)^-2 is irrational.
  8. Sum k ≥ 0 (2k+3)^-2 is irrational.
  9. Sum k ≥ 0 (2k+5)^-2 is irrational.
  10. (1/pi) arccos(1/sqrt(2k+11)) is irrational for k ≥ 0.
• Chapter 6 Quiz
  1. Show sqrt(2) is irrational.
  2. Show x^2 is rational implies x is rational, but that the converse is false.
  3. Show that if is e^s irrational for all integers s, then e^q is irrational for all rationals q.
  4. Show int_( [0,1]x[0,1] ) 1/(1-xy) dx dy = zeta(2) by expanding 1/(1-xy) as a geometric series.
  5. Show zeta(1) diverges.
  6. Show the non-trivial zeroes of the zeta function lie on the line Re(z)=1/2.
• Chapter 10 Quiz
  1. Define: plane graph, spanning tree, dual graph.
  2. Give examples illustrating why "connected" and "plane" are necessary in the statement of Euler's formula.
  3. Compute N-E+F for a graph that lies on the surface of a shpere, a torus, and a genus two (two holes) surface. Conjecture something.
  4. Briefly explain why K_5 is not planar
  5. Show every elementary triangle (i.e., integer vertices, no interior integral points) has area 1/2.
  6. Draw a picture to explain what the Sylvester-Gallai theorem is saying.
• Chapter 13 Quiz
  1. Relate obtuse angles to a vector dot product
  2. Describe conv(S) where S = { (0,-1),(0,0),(1,0),(0,1))
  3. What is the dimension of a hyperplane in R^6?
  4. Show that if Q is a d-polytope, then Q* = {(x-y)/2 : x,y in Q} is a d-dimensional, convex, and centrally symmetric polytope.
  5. Let N = max # { S susbset R^d | the translates Q+s_i of some d-dimensional convex polytope Q subset R^d touch pairwise}. Explain why N is unchanged if Q is replaced by Q* = {(x-y)/2 : x,y in Q}.
  6. State and prove Markov's inequality.
• Chapter 13 Quiz
  1. Give definition of:
     1. convex hull
     2. convex polytope
     3. d-simplex
     4. two convex sets touch
  2. Show that for every d: 2^d ≤ max#{S subset of R^d| angle(s_i,s_j,s_k) is not obtuse for any s_i,s_j and s_k in S} ≤ max#{S subset of R^d| for any two points s_i and s_j in S there is a strip S(i,j) that contains S, with s_i and s_j lying in the parallel boundary hyperplanes of S(i,j)}
• Chapter 13 Quiz
  1. According to Theorem 2, how large is S in R^3? What is the smallest n such that S contains an "angle" in the nontrivial sense?
  2. What does it mean for a set to be centrally symmetric?
  3. Prove rigorously that the property of containment in a half-space is preserved under translation.
  4. Define "linearity of expectation".
  5. Summarize the proof of Theorem 2. Explain why the probability of a bad triple is (3/4)^d in R^d.
• Chapter 22 Quiz
  1. What is an antichain?
  2. Give an example of an antichain of size 10 in the case when N={1,2,3,4,5}.
  3. How many chains of subsets of N={1,2,3,...,n} are there?
  4. What is the size of the largest antichain of the set N={1,2,3,4,5,6,7,8}?
  5. What is an intersecting family?
  6. Prove that the size of the largest intersecting family of N={1,2,3,...,n} is 2^{n-1}.
  7. What is the size of the largest intersecting 3-family in the set N={1,2,3,4,5,6,7}?
  8. Give the definition of a system of distinct representatives.
  9. Is there a system of distinct representatives for a collection of |X|+1 subsets of the finite set X?
  10. Let A_1,...,A_n be a collection of subsets of a finite set X. If there is a system of distinct representatives, then the union of any m sets A_i contains at least m elements (m=1,...,n). Prove.
• Chapter 32 Quiz (see also ch13)
  1. Give definition of:
     1. probability space
     2. random variable
     3. expectation
     4. family of sets is 2-colorable
  2. Show that (2m)^2 x (3/4)^d is less or equal to 1, where m=floor(1/2 x (2/sqrt(3))^d).
  3. Show that the family of all d-subsets of a given (2d-1)-set X is not 2-colorable.
  4. Every family of at most 2^{d-1} d-sets is 2-colorable. Prove.
• Cardan's Formula
  1. Graph x^3+px-q for positive p and q.
  2. Suppose x^3+px=q. If u^3+v^3 = 1 and 3uv+p = 0, then x=u+v is a solution.
  3. Solve the system above to derive Cardan's formula x=cuberoot(q/2 + sqrt(q^2/4+p^3/27))-cuberoot(-q/2 + sqrt(q^2/4+p^3/27))
  4. Explain why this must be real.
  5. Reduce x^3+ax^2+b^x+c by substituting x = y - a/3. What are p and q?
  6. Use the formula to solve
     x^3+x=11
     x^3-5x=100
     x^3-6x^2+11x-6=0
  7. Use the formula to solve x^3=15x+4. Show 4 is a solution. Explain how the formula's answer must be 4. Show cuberoot(2+sqrt(-12))-cuberoot(-2+sqrt(-12))=4.