Due date:

Tuesday, January 31, 2023 5:30 pm

Assignment description

Submission Instructions

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Please note that this assignment is due on Tuesday before the beginning of class

Pre-reading

We will spend the next two weeks (through February 9) discussing Sets, Functions, and Sums. Our discussion will involve all of Chapter 2 except section 2.6 (which you will encounter in a linear algebra course), and also touch on section 3.2. Please read sections 2.1--2.4 to prepare this week.

• {Section 2.1 (Sets) The fundamental question associated with a set is which objects are its elements. Pay attention to examples so that you become familiar with the notation and terminology of sets. You should be prepared to see and interpret symbols like $x∈S, A⊂B,|S|,A×B,P(A), and A∖B$. You should try to understand the notation used in the roster method before we start our in-class discussion, but set-builder notation will be developed as we need it.
• Section 2.2 (Set Operations). When you read this section, look for parallels to our discussion of propositional logic. The basic set operations of Union, Intersection, and Complementation are analogous to the logical operations of Disjunction, Conjunction, and Negation, and the table of set identities (Table 1) should remind you of the table of Logical Equivalences (Table 6 of section 1.3.2). Other operations from propositional logic also have analogues in set theory. Look at Example 13, and compare the method of membership tables to the use of proof tables, and think about how the statement $A⊂B$ is analogous to the statement $p→q$.
• Section 2.3 (Functions). A functions pairs each element of one set, the domain, with an element of a second set, the codomain. Our study of functions will have two main themes.
  • The ideas of surjection and injection are closely tied to the two principles of counting: if you want to count things correctly, you need to count everything at least once, and you cannot count anything more than once. A function with both properties is called a bijection. These ideas will be essential to understanding section 2.5 next week.
  • We will also discuss how addition, multiplication, composition, and inversion interact with functions, and look at some specific functions, floor, ceiling, and factorial functions, that you might not have encountered in high school. The arithmetic of functions is needed for section 3.2.
• Section 2.4 (Sequences and Summations). Functions with domain $\N$ or $\Z+$ are special enough that we often using a different notation for dealing with them. We will refer to such a function as a sequence, and will examine several common kinds of structured sequences, including arithmetic sequences and geometric sequences. We will revisit many examples of this section as we learn about induction (Chapter 5) and recurrence relations (Chapter 8) later in the course.

Re-Reading

We have finished our introductory discussion to logic, corresponding to Chapter 1 of the textbook. Many of the ideas we started exploring in that chapter, particularly proof techniques, will be revisited as we encounter applications throughout the rest of the course.

This would be a good time to review your list of definitions from Chapter 1. For each definition, try to test your understanding by looking for an example of an object that satisfies the definition, and example of an object that doesn't satisfy the definition, and an example of an object that you are not sure about.

A1 (2 points)

You started building a list of theorems and results as part of assignment 1. How many entries are there on your list now?

Describe a result on your list that would have surprised you when you were in grade 10.