Name:______

  1. Let us denote by L the set of all leaves in the world at this point in time, by S the set of all sand grains in the world at this point in time, by P the set of all pennies in the world at this point in time, and by N the set of natural numbers. You are given a 1 yard long stripe on which you are asked to place symbols for these sets in a way that reflects their sizes: that is, smaller sets should be closer to the left end than larger sets, and the distance between the positions of two consecutive sets should be proportional with the difference in size between the sets. Draw a sketch of what your placement of these symbols would be, and explain your reasoning.
  1. 0.9999… (0 followed by an infinite number of 9’s after the decimal point) is

a) less than 1

b) equal to 1

c) greater than 1

d) the last number before 1

e) almost equal to 1

f) cannot be compared to 1

Check all that apply and explain your answer.

  1. Jane said that 1/3 is a “better way” of expressing 0.3333… because 0.3333… is not a precise quantity, as it “never ends”. Do you agree with Jane? Explain your answer.
  1. Suppose you are given an infinite set of numbered tennis balls and two bins of unlimited capacity. Imagine that you place balls 1 and 2 in the first bin and then immediately move ball 1 to the second bin. Then you place balls 3 and 4 in the first bin and move ball 2 to the second bin. This process is continued ad infinitum. What are the contents of the two bins when this process is finished?
  1. The power set of a set A is the set of all subsets of A. For example, if

A = {1, 2, 3}, then its power set is

P(A) = {∅, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}.

Let N denote the set {1, 2, 3, …, n,…}, and P(N) denote the power set of N. Prove or disprove the following statement:

(provide arguments for all your statements)

  1. Consider segment AB and an arbitrary point C on it (see figure below). We divide segment AB into two equal parts, denoting its midpoint by H. Then we divide each of the segments AH and HC into halves, denoting their respective midpoints by P and Q. Imagine continuing to divide segments in this manner. Assuming that all of these steps have been performed, is it necessarily true that C must coincide with one of the midpoints constructed during the aforementioned process?
  1. Here are two definitions for “parallel lines” in plane geometry:

1. Two lines in a plane that never intersect are called parallel lines.

2. Two lines in a plane that do not intersect are called parallel lines.

Do you have a preference for one of these definitions, or do you consider them equivalent? Explain your answer.