Math 13, Winter 2014

Begin Ch. 2 (Logic)
Jan 13: 2.1–2.3. Exercises 2.6ab, 2.14ac, 2.15, 2.17. (For Exercise 2.14 you should state the negations in a natural way, rather than writing "it is not the case that....")
Jan 15: 2.4–2.6. Exercises 2.20, 2.25ab, 2.32b, 2.38def. (For Exercise 2.25 the wording may be confusing. For part (a), assume the statement labeled (a) is true and determine whether the "if...then..." statement is true or false. For part (b), assume the statement labeled (b) is true and determine whether the "if...then..." statement is true or false.)
Jan 17: 2.7–2.10. Exercises 2.48, 2.52a, 2.59, 2.79 (For Exercise 2.59 use De Morgan's laws to write the negation of a disjunction as a conjunction and vice versa. For Exercise 2.79 it may help to first consider the open sentence $\forall a \in A, Q(a,b)$. You can call this open sentence $R(b)$ and investigate the truth of $R(b)$ for various $b \in B$ in order to determine whether the statement $\exists b \in B,R(b)$ is true.

Jan 20: Holiday (MLK Jr. Day)
Begin Ch. 3 (Direct Proof and Proof by Contrapositive)
Jan 22: 3.1–3.2. Exercises 3.1, 3.5, 3.8, 3.11. (Hint for Exercise 3.1: Complete the square.)
Jan 24: 3.3–3.4. Exercises 3.16, 3.21, 3.26, 3.28. (Hint for Exercise 3.28: Prove the contrapositive. Assume that $x$ and $y$ are not both odd, and consider the cases that naturally arise.)

Begin Ch. 4 (More on Direct Proof and Proof by Contrapositive)
Jan 27: 4.1. Exercises 4.4, 4.11.
Jan 29: 4.2–4.3. Exercises 4.16, 4.30, 4.32
Jan 31: 4.4–4.6. Exercises 4.42, 4.53, 4.64. (For Exercise 4.64, your necessary and sufficient condition should not involve "$\times$".)

Begin Ch. 5 (Existence and Proof by Contradiction)
Feb 3: 5.1–5.2. Exercises 5.4, 5.17, 5.25, 5.26
Feb 5: 5.2–5.5. Exercises 5.27, 5.43, 5.48, 5.50. (For Exercise 5.48, see the Intermediate Value Theorem on page 134. The theorem applies to real numbers, although this is not stated explicitly.)
Feb 7: Review

Feb 10: Midterm exam
Begin Ch. 6 (Mathematical Induction)
Feb 12: 6.1. Exercises 6.1, 6.2, 6.3.
Feb 14: 6.1–6.2. Exercises 6.4, 6.10, 6.25. For Exercise 6.4(1) this related picture may be useful or at least interesting. For Exercise 6.25 note that the induction step consists of proving a statement of the form "for all integers $k \ge 4$, if $P(k)$ then $P(k+1)$." That is, to prove $P(k+1)$ you may use the assumption $k \ge 4$ as well as the induction hypothesis $P(k)$.

Feb 17: Holiday (Presidents' Day)
Feb 19: 6.3–6.4. Exercises 6.35, 6.40, 6.42, 6.45. For Exercise 6.40 it might help to think of $S$ as a set of coins: if I have a 1 cent coin, a 2 cent coin, a 4 cent coin, an 8 cent coin, etc. then I can pay any amount $n$ with exact change. (One could also make an analogy with the binary representation of numbers by a computer.) In Exercise 6.42 a "formula for $a_n$" means an equation of the form $a_n = f(n)$ where the expression $f(n)$ depends only on $n$ and not on the terms $a_1,a_2,a_3,\ldots$ of the sequence—such an expression should be easy to guess after you write out the first several terms of the sequence. For Exercise 6.45, note that for the first few values of $n$ you will not need to use the induction hypothesis but for larger values of $n$ you will need to use it.
Begin Ch. 8 (Equivalence relations)
Feb 21: 8.1–8.3. Exercises 8.1, 8.15, 8.18abc, 8.23.

Feb 24: 8.3–8.4. Exercises 8.26, 8.31, 8.38. For Exercise 8.26, say what $R$ is by listing all of its elements (which are ordered pairs.)
Feb 26: 8.5–8.6. Exercises 8.55d, 8.59a, 8.61a. For Exercise 8.61a, prove that your answer is correct.
Begin Ch. 9 (Functions)
Feb 28: 9.1. Exercises 9.2, 9.12b. For Exercise 9.12b, this example solution to 9.12a might help you see how to use the definition of "image".

Mar 3: 9.2–9.4. Exercises 9.26bc, 9.29, 9.32
Mar 5: 9.5–9.6. Exercises 9.42d, 9.53, 9.59
Begin Ch. 10 (Cardinalities of Sets)
Mar 7: 10.1. Exercise: Prove that if $A$ and $B$ are disjoint finite sets, then $A \cup B$ is finite. (A set is called "finite" if it is empty or there is a bijection between it and the set $\{1,2,\ldots,m\}$ for some $m \in \mathbb{N}$. You are asked to prove the statement in terms of this definition, not in terms of any intuition you may have about finite sets. In particular, arguments in terms of "number of elements" are unlikely to constitute a proof—you need to demonstrate the existence of a bijection. By the way, it is also possible to prove the statement without assuming that $A$ and $B$ are disjoint, but this is slightly harder.)