For numerical answers, if it is finite, give a number. If it is infinite, state whether is is countable (has the same cardinality as the integers), or uncountable (which we define here as "the same cardinality as the reals").

You are welcome to use online resources to learn or remind yourself what are equivalence relations and equivalence classes or what steps you must undertake to prove a relation has some property. I've looked, and don't believe that any problem below has an answer online, but if you do find such a solution, copy it, and turn it in, that would constitute a violation of academic integrity.

Problems:

1. You reach, blindly, into a bin with 2 red m&m's, 5 green m&m's, and
    7 blue m&m's.  

  a) You pull out 2 m&m's.  What is the probability they are the same color?
  b) How many must you take out to be sure that you have at least 3
     colors?
  c) if a "handful" of m&m's depends only what colors the handful has,
     and not, for instance, what order they were picked on or their
     arrangement on your palm, how many different handfuls can you
     grab from the bin?
  d) Generalize your answer to (c) for the case where there are "r"
     red m&m's, "g" green m&m's, and "b" blue m&m's.
  e) Recalling that the "redness" in red m&m's used to be
     carcinogenic, write an essay on the trilateral symbolism of
     chocolate as (i) a tasty treat, (ii) a mild narcotic, and (iii) 
     good turkey-stuffing (0 points).

2. Let A be the set {1, 2, 3, 4, 5, 6, 7, 8, 9}

  Let R be the relation { (1,2), (1,1), (3,4), (4,5), (3,9), (6,9),
  (7,8) }

  a) Is R: reflexive?  anti-reflexive?  symmetric? anti-symmetric?

  b) List the pairs that are in the transitive closure of R that are
  not in R.

3. we define the relation ~ on the set of integers to be x ~ y if
        x2 = y2.

  a) (1,1) is an element of this relation, as is (-1,1).  List at
      least 5 other elements of this relation.
  b) Describe the set of all equivalence classes for this relation.
      How many are there?  

4. Prove: For any 5 points on the plane, where each point has integer
   coordinates, that the midpoint of at least one of the lines
   connecting pairs of points also has integer coordinates.

5. Let R be any symmetric relation on set A.  Show that Rn
   is symmetric.

6. Let R be a relation on Real numbers such that 
       R = { (a,b) |  for some k, a = b + 2 * k * pi }.
 show that R is an equivalence relation.
 list 3 elements in the equivalence group [ pi / 2 ]
 How many equivalence classes are there?