Home Syllabus Schedule Links **Homework** | | | | # Homework ## Homework 11, due Dec 1 at 11am Dumas-McCarthy p121: 4.7, 4.8, 4.11, 4.14 Dumas-McCarthy p146: 5.3, 5.16, 5.19, 5.23 And the following additional problems: **A.** Explain the trouble with the "all chalk is white" proof that was presented in class. (I want to know the problem in the proof, not that there may exist orange or blue chalk.) **B.** The Sandwich Theorem: Prove that if *f*(*x*) ≤ *g*(*x*) ≤ *h*(*x*) for all *x * ∈ **R** and lim_{x→∞} *f*(*x*) = lim_{x→∞} *h*(*x*) = *L*, then lim_{x→∞} *g*(*x*) = *L*. ## Homework 10, due Nov 19 at 11am Dumas-McCarthy p93: 3.25, 3.27, 3.31 Dumas-McCarthy p121: 4.1, 4.2, 4.4, 4.5, 4.6 For problem 4.6, please prove your statement (likely using induction). On a previous homework we'd given an informal argument for this statement, and we're being a little more formal and careful with our proofs at this point in the semester. ## Homework 9, due Nov 12 at 11am Dumas-McCarthy p93: 3.13, 3.14, 3.15, 3.18, 3.19, 3.23, 3.24 ## Homework 8, due Oct 29 at 11am Dumas-McCarthy p93: 3.1, 3.2, 3.4, 3.5, 3.6, 3.7, 3.8 ## Homework 7, due Oct 22 at 11am Dumas-McCarthy p65: 2.5, 2.6, 2.15, 2.16, 2.17, 2.19, 2.20 ## Homework 6, due Oct 15 at 11am Dumas-McCarthy p65: 2.13, 2.21, 2.24, 2.26 **A.** i) Show that if *X* has an equivalence relation ~ on it, and *Y* is a subset of *X*, then the restriction of ~ to *Y* × *Y* is an equivalence relation. ii) Show a similar result for a partial order ≤ on *X*. **B.** Let Π_{n} be the family of all partitions of {0, 1, 2, ..., *n* - 1}. There is a partial order defined on Π_{n} by *x* ≤ *y* if every part of *x* is contained in some part of *y*. Draw the diagram discussed in class (and at the bottom of Chapter 2.2) for Π_{4} with this order. How many partitions are there in Π_{4}? ## Homework 5, due Oct 8 at 11am Dumas-McCarthy p65: 2.3, 2.4, 2.7 (prove your answer is correct), 2.12, 2.22 **A.** Let *X*_{1} and *X*_{2} be sets. Let *X*_{1} have a partial order *≤*_{1} and *X*_{2} have a partial order *≤*_{2}. A function *f* : *X*_{1} → *X*_{2} is said to be *order-preserving* if whenever we have *x* *≤*_{1} *y* (for *x*, *y* in *X*_{1}), then we also have *f*( *x* ) *≤*_{2} *f*( *y* ). If *X*_{1} is the power-set on {0, 1} ordered by inclusion, and *X*_{2} is the linearly ordered set {0 < 1 < 2 < 3}, then how many order-preserving maps are there from *X*_{1} to *X*_{2}? ## Homework 4, due Oct 1 at 11am Dumas-McCarthy p41: 1.20, 1.27 Dumas-McCarthy p65: 2.2, 2.8, 2.9, 2.10, 2.11 **Note:** *y* should be *Y* in 1.27. Also, the symbol | means "divides", and you can find a careful definition of what this means as the very first definition of Section 2.5, at the top of p61. ## Homework 3, due Sep 10 at 11am Dumas-McCarthy p41: 1.9, 1.10, 1.11, 1.12, 1.15, 1.17 ## Homework 2, due Sep 3 at 11am Dumas-McCarthy p41: 1.4, 1.5, 1.7, 1.8, 1.13, 1.16 **Note:** The homework uses some definitions that we have not yet seen in lecture, although not in a difficult way. The definition of an interval in **Z** may be found at the beginning of 1.6. Injections, surjections, and bijections are defined in 1.4. ## Homework 1, due Aug 27 at 11am Read Chapter 0, and the rest of Chapter 1.2.2. Then do Dumas-McCarthy p41: 1.1, 1.2, 1.3 And the following additional problems: **A.** Let *a*, *b*, *c*, and *d* be in ℝ (the reals) with ~~a < ~~*b*, *a* < *c* < *b* < *d*, ~~and ~~*c* < *d*. By showing that each side is a subset of the other, show that [*a*, *b*] ∪ [*c*, *d*] = [*a*, *d*]. **B.** If *a*, *b*, and *c* are in ℝ (the reals) with a < *b* < *c*, then show that [*a*, *b*] ∪ [*b*, *c*] = [*a*, *c*], but that (*a*, *b*) ∪ (*b*, *c*) is a proper subset of (*a*, *c*). |