Section 1.1 Read and Study the Section. Pay Particular Attention to the Following

Math 141Spring 2018

Prep for Wednesday, 1/10/18

Section 1.1 –Read and study the section. Pay particular attention to the following.

a)What are the 3 main categories for solving problems with a finite number of possibilities? Give an example of each different from the text. Group members should use different examples.

b)What is an algorithm?

c)Read the PERT process carefully so that you understand the examples.

------

Math 141Section 1.1 Reading Quiz – Short Answer1/10/18Name______

What are the 3 main categories for solving problems with a finite number of possibilities?
Briefly give an example of an existence problem different from the text.
What is an algorithm?
What is a critical path in the PERT method?

Math 141 Group Work and Discussion – January 10th.

You are to make sure that each group member understands the PERT method and can implement it. You first work problems where you use the method on problems given the diagram to determine total project time and critical paths. The next step is to start with a table to construct the diagram and finish the problem as in the previous problem. The third type problem is to start with a verbal description of the problem and continue until you determine total project time and critical paths.

Work problems 7, 11, 17, 18 so that we have a common set of problems to discuss. Submit one solution per group. (This can be done today, or Friday, or next Wednesday since I am out of town.) As time allows select additional problems or work them outside of class so that you have mastered the technique and can perform it on a quiz.

Use each other as resources. If questions arise this week that need more input from me, send me an e-mail regarding the question or topic. I will address it next week.

The following statement in Friday’s assignment refers to the Algorithm for Evaluating xn and Polynomial Evaluation Algorithm. The text has an example of implementing Horner’s Algorithm for a third degree polynomial.

Step through the algorithms for n = 5. Record these steps.

Prep for Friday, 1/19/18

Section 1.2 –Read and study the section. Pay particular attention to the following.

a)What does n! represent?

b)How many ways are there to order 5 items?

c)What does factorial have to do with the pilot/flight example?

Section 1.4 – Read and study the sections Comparing Algorithms & Evaluating Polynomials. Pay particular attention to the following.

a)Step through the algorithms for n = 5. Record these steps.

b)What is the general concept of the complexity of an algorithm?

c)Why does complexity matter? Consider examples earlier in the chapter.

------

Math 141Section 1.1 Reading Quiz – Short Answer1/19/18Name______

What are the 3 main categories for solving problems with a finite number of possibilities?
Briefly give an example of an existence problem different from the text.
What is an algorithm?
What is a critical path in the PERT method?

Math 141 Group Work and Discussion – January 19th.

Compute by hand # 2, 3, 6, 7, & 12. You can check these with a calculator if you need to do so. You do not need to submit these problems. Some may have experience working with factorials. Make sure that everyone in the group understands how to simplify them.

Under the conditions on page 15, how long would it take to check all possible matchings of 10 pilots and 10 flights? Show your work. Is this practical?

Work #10, discussion of how you stepped through the algorithms in preparation for today may help.

Compose a group answer for what complexity of an algorithm is and why it matters.

Turn in group solutions to #2-4 by Wednesday.

Prep for Monday, 1/22/18

Section 2.1 –Read and study the section. Pay particular attention to the following.

Understand union, intersection, complement.
Understand the notation:
Be able to describe in words (and symbols are okay too) the commutative, associative, distributive laws
Understand how the Venn Diagrams on page 44 depict the sets.
Be able to state De Morgan’s Laws – at least write be able to write down the symbols in (a) & (b) of Theorem 2.2.

------

Math 141Sections 1.2 & 1.4 Reading Quiz – Short AnswerName______

What does 8! represent?

What does factorial have to do with the pilot/flight example?

Group Work Monday, January 22, 2018

What set operations are discussed in thm 2.1?

Consider the first 3 laws as they relate to the operations of arithmetic and algebra.

Can you see pictures (Venn diagrams) for the rest of the statements in thm 2.1? Have different members of the group sketch Venn digrams of the statements and share the results.

Have each member sketch a Venn diagram of a side of an equation in Thm. 2.2. Read through the proof. We’ll go through it together.

What have you worked with that is a Cartesian product? Create 2 sets whose elements are not numbers and list some Cartesian products.

Exercises from book
#1, rewrite 9, 10. Venn diagrams for 9-11
13, 15, 26-28
20, 21, 22, 23, 24, 19

------

Section 2.1 Reading Quiz – Short Answer2/24/18Name______

Write down in words what each of the following denotes.

Draw a Venn diagram for each of the following.

Explain in words what the Communitive law of sets means.

State a version of DeMorgan’s Laws.

Math 141 Modified Preparation for Friday, 1/26/18

Carefully read 2.2 and 2.4. Understand all terms/phrases in bold in these sections. Section 2.4 should be a review. The Reading Quiz will come from these terms/phrases.

Study Theorem 2.3 in light of our discussion justifying Theorem 2.2 a. Continue to develop the idea of justifying statements by filling in the blanks to #32 below.

To see if you understand the concepts in section 2.2 work 1, 2, 5, 19.

Work # 19 to verify that you understand the concept.

Page 47 #32. Let A and B be subsets of a universal set U. Show that if, then .

Let

Our assumption is;

so knowing that , tells us that it cannot be an element of ______.

Restating this we have or .

Therefore ______which is what we wanted to show.

The reasoning behind the justification above is that if you take an arbitrary element of ______, you can use the

assumption (the if statement) to show that the arbitrary element is also ______,

------

Math 141Section 2.3January 26, 2018

Theorem 2.3

Let R be an equivalence relation on a set S.

In I and II, let ,

If xRythen [x] = [y].

Let , then ___R ___

By our assumption ___R___

Since R is an equivalence relation, this implies that ___R___

So .

Thus .

Similarly,

Let , then ___R ___

By our assumption ___R___. Using the symmetric property of R we have ___R___,

Since R is an equivalence relation, this implies that ___R___

So .

Thus .

Since and , we have shown

If [x] = [y], then xRy.

We know that by the ______property.

Sosince ______.

But implies ___R ___, which is our desired conclusion.

Two equivalence classes are either equal or disjoint.

Let [u] and [v] be any two equivalences classes of R.

If [u] and [v] are disjoint then the statement is true.

So let’s consider the case where [u] and [v] are not disjoint.

This means that there is an element, call it w, that is in both _____ and _____.

Since , ____R ____, or

Since , ____R ____, or .

It follows .

So that [u] and [v] are either disjoint or equal which is what we wanted to show.

------

Math 141Section 2.2 & 2.4 1/29 Reading Quiz – Short AnswerName______

Explain the following terms as used in these sections.

Equivalence relation
Partition of a set
One-to-one function
Identity function
Inverse function

Group Work

Sec. 2.2Theorem 2.3

1, 3, 4, 9, 11*,

12, 13*, 14*, 16show that it’s an equivalence relation, describe equivalence class

20,

Give a reasonable explanation for 22

28, 29

Turn in as group with explanation for each 11, 13 as above, 14 describe equivalence class

Sec. 2.41-3, 5, 6, 8, 21-35 odd;

before using a calculator on the second set bound/estimate the answer

61 (note 59)

Collected individual work –Include explanations. No credit for just answers. Due 1/29

1.1#18

1.1#14, 22, 18

1.2#2, 6

Math 141January 29, 2018Name______

Let S be the set of all subsets of {1, 2, 3, 4} and X R Y means that . Determine which of the reflexive, symmetric, transitive properties R has. Explain.

Write the equivalence relation on {1, 2, 3, 4, 5, 6} that is induced by the partition {1,3,6}, {2,5}, and {4} as its partitioning sets.

What is the inductive step in mathematical induction?
If you want to show that for any positive integer n using mathematical induction, what would be the base for induction?

______

Sec. 2.5 Group work –

#11-14

Basis

Assumption

Show next step

State conclusion

Discuss your sketches example 2.52 – is it obvious for a 21 x 21 checkerboard? how could you cover a 22 x22 checkerboard? a 23 x 23 checkerboard?

#7 – don’t look at the back of the book before you have an answer!

______

For Wednesday- recursive definition, Fibonacci numbers, what is the difference between the Principle of Mathematical Induction and the Strong Principle of Mathematical Induction? Study examples 2.53 & 2.54 so that you either have specific questions or could repeat them without the text.

Math 141January 31, 2018Name______

What is the Fibonacci sequence?

What is the difference between the mathematical induction that we discussed Monday and the Strong Principle of Mathematical Induction?

January 31, 2018

By groups put on board #11, 12, 13, choice 16 or 22

*Talk in groups about inductive step for Every set with n elements has 2n subsets.

In section 2.5, you should do those from Monday as well as 16, 19, 22, & *above by mathematical induction.

We’ll discuss checkerboard. #13 sec 2.2

Collected individual homework- Due 2/2 Explanations are necessary. Give solutions, not just answers.

1.1#16 Use PERT!

2.2 #18, 21

2.5# 14

For Friday, 2/2/18 –

Read Appendices A.1 & A.2

Be able to define – statement, conjunction, disjunction, premise, conclusion, biconditional, tautology

Know how to negate statements with “some”, “all”, and “every”.

Know when a conditional statement is true and when it is false.

Statement

Converse

Contrapositive

Inversewhich are logically equivalent

Types of proof – read A.3 to understand what direct proof, proof by contradiction, counterexample to disprove a statement

------

Math 141February 7, 2018Name ______

What is the division algorithm? You may use variables. What are the conditions on the remainder?
Find the quotient and remainder in the division of 67 by 9.
When is ?
If , what must you show to demonstrate that congruence mod m is symmetric? Do it.

2/7 Prep Topics for reading quiz:

Know what the division algorithm is and the conditions on r. Be able to illustrate with #1-3 on pg 105

Define “a is divisible by b”.

Define congruence mod m, i.e. explain x is congruent to y modulo m. (Be able to demonstrate with # 11 & 15.)
Is congruence mod m an equivalence relations? Justify. (Which properties do you need to check?)

Math 141February 7

Groups

Together establish that congruence modulo m is an equivalence relation.

If m = 7, what does [3] look like?

You should be able to easily do 1-36.

1)How do you apply Theorem 3.2? Why would we need this?

2)p. 105 3-8, 9-15 odd

3)Work 9, 19, 21, 22, 24, 25, 27, 30, 32, 33, 35 multiple ways.

For Friday

In addition to finishing the above, carefully reread sec 3.1 with particular attention to applications. Work out the check digit for another of your books & bring the example to class Friday. What is the algorithm for determining the day of the week, and why does that work? Model problem 37 with techniques of this section.

Math 141February 9, 2018Name ______

Find the check digit for the ISBN whose first 9 digits are032130515.
What day of the week will February 9, 2025 be?

Groups:

3.137, 39, Check an ISBN from a text that you have; 41, 43a

Find the remainder modulo 9 & modulo 11 of: 7349621 372342323 138030695

How can you tell if a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 11?

Proving the algorithm for 7 takes more time and I’ll do it Monday.

Can you show why the algorithm works for 3?

For Monday be able to work 1 – 41 sec. 3.1; you’ll work on 44-47. Review for Test Friday.

No reading quiz 2/12 & 14. 2/12 went over algorithms for divisibility 2/14 worked on review sheet for test.

Math 141 Test 1 NotesSpring 2018

There are some suggested problems from summaries. More practice on that type, or a type problem that wasn’t in the summaries are listed in the notes below. You will not have a proof (induction & A.3) on the test that isn’t specifically listed here.