Homework
Discrete Math, MATH 2001-001, Fall 2017
(Click here to return to the MATH-2001 course website.)
Information on working together on daily homework:
Unless otherwise noted you may work together on the daily homework, to help each other understand and solve the problems. The write-up of the solutions, however, should be done independently and should be your own work. "Your own work" means that during the write-up you are working alone and using your own
words. It means that if you were to re-do the assignment at a later date, the
result would be of similar style, quality and depth of understanding.
Do the first problem on this sheet. On a separate sheet of paper,
do your best to solve the second one as well (it will be due Wednesday).
RecursionAndInductionPractice
Read and study Chapter 10, through page 164.
The exam is next Wednesday. Study suggestions are:
Complete and study Counting worksheets 1 and 2 (solutions posted).
Practice proofs by contradiction and by induction.
There are many counting problems in the textbook for you to practice with. Odd numbers have solutions in the back of the textbook.
Come to class with any questions you can't resolve.
There is nothing to turn in on Monday.
Complete Counting Worksheet 2
Do problem 23 from page 170 (to turn in). The problem is to prove the binomial theorem.
Practice for the quiz by doing three proofs by induction from the text. Do one that involves a summation (Chapter 10, problems 1-8, 15, 20), one that involves an inequality (16, 19, 21, 22), and one that involves divisibility (9-14).
divisibility. You don't need to turn these in. You can consult the solutions to odd problems at the end of the textbook as models.
Complete the first page of Counting Worksheet 2 that we began in class.
Choose your weakest technique of proof (truth tables, direct, contrapositive, contradiction, or proof by induction) and do one proof from that technique (from textbook).
Continue reading Chapter 10, up to but not including section 10.1
Do Chapter 10 number 4
And prove that n! > 2^n for natural numbers n \geq 4.
Based on ideas and strategies discussed in class, redo the proof that sqrt(3) is irrational (or sqrt(5) is irrational). Also redo the proof that the product of three consecutive integers is divisible by 6
Write down at least one question you have regarding the first three pages of Chapter 10.
Choose one of the following:
1. Prove that the square root of 3 is irrational.
2. Prove that the square root of 5 is irrational.
(Which is harder, and why?)
Then read the first three pages of Chapter 10.
Suppose a and b are real numbers. Prove that if a is a rational number and a+b is an irrational number, then b is an irrational number.
Do Chapter 6 problem 8
Prove that the product of three consecutive integers is divisible by 6.
- Do Chapter 5, problems 20, 24
- Then determine if the following statement is true. If it is not true, find
a counter-example. If it is true, prove it:
An integer n is a multiple of 3 if and only if n^2 is a multiple of 3.''
- Then read the introduction to Chapter 6, and sections 6.1 and 6.2.
Do the following problems:
Chapter 4: 19, 28
Chapter 5: 12
Do the following problems:
Chapter 4: 8, 10
Then read Sections 5.1 and 5.2.
And do Chapter 5: 2, 6
Read the remainder of Chapter 4.
Then do the following problems.
Chapter 4: 6, 14
Read pages 87-89 (the first three pages of Chapter 4) and section 4.3.
Then do the following problems.
Chapter 4: 1, 2, 4
You can watch this video after you complete problem 1:
https://www.youtube.com/watch?v=6NKa-iwBJFc
You should already have completed reading Chapter 2. Do the following problems. You do not need to explain yourself on these - just give answers.
Section 2.9: 2, 3, 4, 5, 7, 8, 12
Section 2.10: 1, 2, 3, 4, 6, 7, 11
Complete the Truth Tables Worksheet from class today. You may work together with your group members if you will be meeting with them. You do not need to turn this in.
Finish reading the rest of Chapter 2.
Read sections 2.7, 2.8, and 2.9
Then do these problems (and turn them in):
Section 2.7: 2, 8, 9, 10
Read sections 2.5 and 2.6.
Then do these problems:
Section 2.5: 8
Section 2.6: A2, A8, B10
Read the introduction to Chapter 2, and sections 2.1 and 2.2 from the textbook
Then do these problems:
Section 2.1: evens. Follow the instructions but then also determine whether or not each
example is an open sentence.
Section 2.2: 6, 8
Read section 1.2 from the textbook (on cartesian products)
Then do these problems:
1.2.A.2 (fgh)
1.2.A.4
1.2.A.7
1.2.B.20
- Read sections 1.6 and 1.7 from the textbook
- Do these four problems from the textbook: 1.6.2, 1.6.6 (I suggest you draw two separate graphs), 1.7.6, 1.7.12
Please do the following before class on Monday.
- Read sections 1.4, 1.5 and 1.6. Since we skipped sections 1.2 for the time being, you haven't yet read about Cartesian products. So in the reading for Friday, ignore any examples with Cartesian products in them. (You'll recognize them because they have an "x" in them, for example AxB.)
- Write down 2-3 key concepts from each section.
- Think about these questions (you do not need to write these down, but be prepared to discuss them)
- Why might mathematicians have chose the term "power set"?
- If |A| = n, remind yourself what the formula is for the size of the power set of A.
- Why is this the correct formula?
- Draw Venn diagrams for A intersect B, A union B, and A-B. (If you don't know what a Venn diagram is, then research it)
- Say that A and B are sets, and neither are empty. Answer "Always", "Sometimes", or "Never" for each of the following. If you answer "sometimes", then create examples supporting your conclusion. If you answer "Always" or "Never", then explain.
- A union B is empty.
- A intersect B is empty.
- A-B is empty
- Do these problems:
1.3.A.8
1.3.B.12
1.4.A.2
1.4.B.14
1.5.2
1.5.4f
- Watch this video:
https://www.youtube.com/watch?v=QGnb-ctTQE8
- Read sections 1.1, 1.3 and 5.3 from the text. Take your time.
- Write down 2 differences you notice between how you read a math text and how you read a novel.
- Write down 2-3 key concepts that you got from each section of the reading.
Then solve these problems, showing explanatory work:
1.1.A #8
1.1.B #28
1.1.C #36
1.1.D #46