# Homework

Homework assignments will be posted here, and turned in via Gradescope.

## Homework 1: Due on Friday, January 28 at 11:59 PM ET.

Problems to turn in:
§ 1.3: 3, 4 (for full credit, some explanation is needed)
§ 1.4: 2, 4deg
§ 1.5: 2, 5
§ 1.6: 4, 5
Suggested (do not turn in):
§ 1.3: 2, 6
§ 1.4: 4abcf
§ 1.5: 1, 4
§ 1.6: 1, 3, 6, 8

Solutions

## Homework 2: Due on Friday, February 4 at 11:59 PM ET.

Problems to turn in:
§ 2.1: 1bcd
§ 2.2: 3 (see hints following Theorem 2.3 and table of notation following preface), 6, 7 (use 6), 9, 10

Note: In the Section 2.2 problems, equivalent means equipotent, and you can find a definition of “card A” in the text’s definition equipotence, p.34.

Suggested (do not turn in):
§ 2.1: 2, 3
§ 2.2: 1, 5, 8

Solutions

## Homework 3: Due on Friday, Feburary 11 at 11:59 PM ET.

Problems to turn in:
§ 2.3: 1, 2, 5, 6, 7 (see p.40 for definition of D(A))
§ 2.5: 2 (only Theorem 2.7. Think carefully about what replaces an open interval, etc.).
Suggested (do not turn in)
§ 2.3: 8, 9
§ 2.4: 1
§ 2.5: 1, 2 (Theorem 2.8), 4

Solutions

## Homework 4: Due on Friday, February 18 at 11:59 PM ET.

Problems to turn in:
§ 3.1: 4bc
§ 3.2: 2, 7, 10, 11, 12, 13
Suggested (do not turn in)
§ 3.1: 1, 2, 3, 7, 8
§ 3.2: 6, 7, 14

Some comments on 3.1.4b: You are welcome to use usual properties of continuous functions: sums and differences of continuous functions are continuous, etc., as well as usual properties of definite integrals. If one is trying to show that if a continuous function $$h(x)$$ satisfies $$\int_a^b |h(x)| dx = 0$$, then $$h(x) = 0$$ for all $$x \in [a,b]$$, then one can suppose that $$h(x_0) = m$$ for some $$x_0 \in [a,b]$$ with $$m \neq 0$$ and derive a contradiction using the continuity of $$h$$ by applying the intermediate value theorem (or, if you’d like, the definition of continuity of a function $$f \colon \mathbb{R} \to \mathbb{R}$$). The details are left to you :)

Solutions

## Homework 5: Due on Friday, February 25 at 11:59 PM ET.

Problems to turn in:
§ 3.3: 2, 3, 9, 11
§ 3.4: 3, 4, 5
Suggested problems (do not turn in):
§ 3.3: 4, 5, 7, 8
§ 3.4: 1, 6

Solutions

## Homework 6: Due on Friday, March 18 at 11:59 PM ET.

Problems to turn in:
§ 3.5: 3 (see § 1.7 for a discussion of equivalence relations, if needed), 6
§ 3.6: 1, 2, 4
§ 4.1: 1, 2, 4

Solutions

## Homework 7: Due on Friday, March 25 at 11:59 PM ET.

Problems to turn in:
§ 4.1: 8, 9, 10
§ 4.2: 4 (do some explaining!), 6, 7, 9

Solutions

## Homework 8: Due on Friday, April 1 at 11:59 PM ET.

Problems to turn in:
§ 4.3: 4, 6, 7, 8 (you can use Theorem 4.11 if you need), 10
§ 4.4: 10
Suggested problems:
§ 4.3: 1, 3, 5

Solutions

## Homework 9: Due on Friday, April 8 at 11:59 PM ET.

Problems to turn in:
§ 4.2: 10
§ 4.4: 3, 6, 9
§ 4.5: 2, 3, 10, 14 (only for n = 2)
Suggested problems:
§ 4.5: 1, 8, 9, 11, 12, 13, 15

Solutions

## Homework 10: Due on Friday, April 15 at 11:59 PM ET.

Problems to turn in:
§ 5.1: 4
§ 5.2: 1, 6, 8, 9a, 10, 12, 13 (see paragraph under the second Corollary on p. 135 for the definition of a continuous invariant)
Suggested problems:
§ 5.2: 5, 7, 11, 14, 15, 16

Solutions

## Homework 11: Due on Wednesday, April 27 at 11:59 PM ET.

Problems to turn in:
§ 5.2: 14
§ 5.5: 6, 7, 10
§ 6.1: 3, 4, 5, 6

Solutions