**Due: May 8, 2017 at 14:00.**

*Read and understand the following instructions on submission of
homework.* **If you do not follow them, you will not receive credit.**

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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 |

For any set A:

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

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.

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.

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

- A ∪ B
- A ∩ B
- A \ B
- B \ A

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

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)?

- E ∩ F
- E \ F

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

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

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).

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