Homework #2: Sets

Due: May 8, 2017 at 14:00.

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Problems

Recall that the defining characteristic of a set is that whether any given thing is a member of a set is known, either it is a member or it isn't. It follows that two sets A and B are the same exactly when each member of A is a member of B, and each member of B is a member of A.

The following operations are defined on sets A and B:

A ∪ B union the set of things that are members of at least one of A or B
A ∩ B intersection the set of things that are members of both A and B
A \ B difference the set of things that are members of A but not B

Problem 1

For any set A:

  1. What is A ∪ {}? Explain why in terms of the membership relationship.
  2. What is A ∩ {}? Explain why in terms of the membership relationship.

Problem 2

Consider the following two descriptions of collections:

  • The collection of students in this class who understand mathematics well.
  • The collection of students in this class who get the grade of A.

Which of these describes a well-defined set? Explain why you chose the one you did, and why you rejected the other one.

Problem 3

Consider the two sets containing one member: A = { a professor named Stephen Turnbull } and B = { a professor named Stephen Turnbull }. Explain why you cannot say A = B.

Hint: http://www.amazon.com/Stephen-Turnbull/e/B001H6U2BS.

Problem 4

Let A = { 1, 2, 3 } and B = { 2, 4 }. Compute the following sets:

  1. A ∪ B
  2. A ∩ B
  3. A \ B
  4. B \ A

Problem 5

Consider two sets C and D. What can you say about (C ∩ D) ∪ (C \ D)? Explain.

Problem 6

Let E = { stocks listed on the first section of the Tokyo stock exchange whose price was at least as high on June 30, 2015 as on December 31, 2014 } and F = { stocks listed on the first section of the Tokyo Stock Exchange which traded more shares January 1 -- June 30, 2015 than July 1 -- December 31, 2014 }. What are the following sets (described in the same style as E and F)?

  1. E ∩ F
  2. E \ F

Problem 7

Let G = { ordered pairs of 6-sided dice whose sum is 4 } and H = { ordered pairs of 6-sided dice whose product is 4 }.

  1. Describe G ∩ H in words, in the same style as G and H were defined.
  2. Compute G ∪ H as an explicit set of ordered pairs (x, y) of numbers between 1 and 6 (inclusive).

Problem 8

In the Linda problem, Linda was described as "31 years old, single, outspoken, and very bright." Suppose you pick a 31-year-old woman at random. She can be married or single (= not married), outspoken (= not quiet) or quiet, and very bright or normal (= not outstandingly intelligent).

  1. Why did I define single, outspoken, and normal? (The same answer applies to all three attributes of Linda.)
  2. Using the set operations of union, intersection, and difference, construct as many different events as possible from the properties single, outspoken, and very bright.
  3. How many different events are possible based on those three possibilities?