LaTeX-ed solutions are encouraged and appreciated.
If you use LaTeX, hand-in a printed version of your homework.
You are encouraged to discuss homework problems with classmates, but such discussions should NOT include the exchange of any written material.
Writing of homework problems should be done on an individual basis.
References to results from the textbook and/or class notes should be included.
The following lists should be considered partial and tentative lists until the word complete appears next to it.
Use 8.5in x 11in paper with smooth borders. Write your name on top of each page. Staple all pages.
Problem Set 12 (complete) Due: 12/08/2017. Board presentation: 12/08/2017
Prove that if A and B are finite sets, then A∪B is a finite set.
Prove the following corollary to Proposition 13.6.
If f:A→B is injective and B is finite, then A is finite.
If g:A→B is surjective and A is finite, then B is finite.
Do Project 13.15, finding a formula for the bijection in the picture.
Prove Theorem 13.28.
Problem Set 11 (complete) Due: 12/01/2017. Board Presentation: 12/01/2017
Write down the details of the proofs that the sum of a rational number and an irrational number is irrational, and that the product of a non-zero rational number and an irrational number is irrational.
Prove the converse of Prop. 11.2
Do Project 11.14
Prove that for all x,y,z,w∈R with z,w≠0, xz+yw=xw+yzzwandxzyw=xyzw
Consider the set A={x∈Q∣x2<2} Show that A is non-empty and has an upper bound in Q, but does not have a least upper bound in Q. Hint: by way of contradiction, assume A has a least upper bound u in Q, and compare it with √2.
Consider the sequence defined recursively by an=an−1+3an−2a1=1a2=2. Use the converse of Proposition 11.25 to find a closed formula for an.