**MTH 221 Complete Class Week 1 – 5 All Assignments and Discussion Questions – A+ Graded Course Material**

**Week 1 Individual Assignment Selected Textbook Exercises**

**Complete **12 questions below by choosing at least four from each section.

· Ch. 1 of *Discrete and Combinatorial Mathematics*

o Supplementary Exercises 1, 2, 7, 8, 9, 10, 15(a), 18, 24, & 25(a & b)

· Ch. 2 of *Discrete and Combinatorial Mathematics*

o Exercise 2.1, problems 2, 3, 10, & 13,

o Exercise 2.2, problems 3, 4, & 17

o Exercise 2.3, problems 1 & 4

o Exercise 2.4, problems 1, 2, & 6

o Exercise 2.5, problems 1, 2, & 4

· Ch. 3 of *Discrete and Combinatorial Mathematics*

o Exercise 3.1, problems 1, 2, 18, & 21

o Exercise 3.2, problems 3 & 8

- Exercise 3.3, problems 1, 2, 4, & 5

**Week 1 DQ 1**

Consider the problem of how to arrange a group of *n* people so each person can shake hands with every other person. How might you organize this process? How many times will each person shake hands with someone else? How many handshakes will occur? How must your method vary according to whether or not *n* is even or odd?

**Week 1 DQ 2**

There is an old joke that goes something like this: “If God is love, love is blind, and Ray Charles is blind, then Ray Charles is God.” Explain, in the terms of first-order logic and predicate calculus, why this reasoning is incorrect.

**Week 1 DQ 3**

There is an old joke, commonly attributed to Groucho Marx, which goes something like this: “I don’t want to belong to any club that will accept people like me as a member.” Does this statement fall under the purview of Russell’s paradox, or is there an easy semantic way out? Look up the term *fuzzy set theory* in a search engine of your choice or the University Library, and see if this theory can offer any insights into this statement

**Week 2 Individual Assignment Selected Textbook Exercises**

**Complete **12 questions below by choosing at least three from each section.

· Ch. 4 of *Discrete and Combinatorial Mathematics*

o Exercise 4.1, problems 4, 7, & 18

o Exercise 4.2, problems 11 & 16

· Ch. 4 of *Discrete and Combinatorial Mathematics*

o Exercise 4.3, problems 4, 5, 10, & 15

o Exercise 4.4, problems 1 & 14

o Exercise 4.5, problems 5 &12

· Ch. 5 of *Discrete and Combinatorial Mathematics*

o Exercise 5.1, problems 5 & 8

o Exercise 5.2, problems 2, 5, 12, & 27(a & b)

o Exercise 5.3, problems 1 & 8

o Exercise 5.4, problems 13 & 14

o Exercise 5.5, problems 2 & 7(a)

o Exercise 5.6, problems 2, 3, 4, & 5

· Ch. 5 of *Discrete and Combinatorial Mathematics*

o Exercise 5.7, problems 1 & 6

- Exercise 5.8, problems 5 & 6

**Week 2 DQ 1**

Describe a situation in your professional or personal life when recursion, or at least the principle of recursion, played a role in accomplishing a task, such as a large chore that could be decomposed into smaller chunks that were easier to handle separately, but still had the semblance of the overall task. Did you track the completion of this task in any way to ensure that no pieces were left undone, much like an algorithm keeps placeholders to trace a way back from a recursive trajectory? If so, how did you do it? If not, why did you not?

**Week 2 DQ 2**

Describe a favorite recreational activity in terms of its iterative components, such as solving a crossword or Sudoku puzzle or playing a game of chess or backgammon. Also, mention any recursive elements that occur.

**Week 2 DQ 3**

Using a search engine of your choice, look up the term *one-way function*. This concept arises in cryptography. Explain this concept in your own words, using the terms learned in Ch. 5 regarding functions and their inverses.

# Week 3 Individual Assignment Selected Textbook Exercises

**Complete **12 questions below by choosing at least four from each section.

· Ch. 7

o Exercise 7.1, problems 5, 6, 9, & 14

o Exercise 7.2, problems 2, 9, &14 (Develop the algorithm only, not the computer code.)

o Exercise 7.3, problems 1, 6, & 19

· Ch. 7

o Exercise 7.4, problems 1, 2, 7, & 8

· Ch. 8

o Exercise 8.1, problems 1, 12, 19, & 20

- Exercise 8.2, problems 4 & 5

**Week 3 DQ 1**

What sort of relation is *friendship*, using the human or sociological meaning of the word? Is it necessarily reflexive, symmetric, antisymmetric, or transitive? Explain why or why not. Can the friendship relation among a finite group of people induce a partial order, such as a set inclusion? Explain why or why not.

**Week 3 DQ 2**

Look up the term *axiom of choice* using the Internet. How does the axiom of choice—whichever form you prefer—overlay the definitions of equivalence relations and partitions you learned in Ch. 7?

**Week 3 DQ 3**

How is the principle of inclusion and exclusion related to the rules for manipulation and simplification of logic predicates you learned in Ch. 2?

# Week 4 Individual Assignment Selected Textbook Exercises

**Complete **12 questions below by choosing at least four from each section.

· Ch. 11 of *Discrete and Combinatorial Mathematics*

o Exercise 11.1, problems 3, 6, 8, 11, 15, & 16

· Ch. 11 of *Discrete and Combinatorial Mathematics*

o Exercise 11.2, problems 1, 6, 12, & 13,

o Exercise 11.3, problems 5, 20, 21, & 22

o Exercise 11.4, problems 14, 17, & 24

o Exercise 11.5, problems 4 & 7

o Exercise 5.6, problems 9 &10

· Ch. 12 of *Discrete and Combinatorial Mathematics*

o Exercise 12.1, problems 2, 6, 7, & 11

o Exercise 12.2, problems 6 & 9

o Exercise 12.3, problems 2 & 3

- Exercise 12.5, problems 3 & 8

**Week 4 DQ 1**

Random graphs are a fascinating subject of applied and theoretical research. These can be generated with a fixed vertex set *V* and edges added to the edge set *E* based on some probability model, such as a coin flip. Speculate on how many connected components a random graph might have if the likelihood of an edge (*v1,v2*) being in the set *E* is 50%. Do you think the number of components would depend on the size of the vertex set *V*? Explain why or why not.

**Week 4 DQ 2**

You are an electrical engineer designing a new integrated circuit involving potentially millions of components. How would you use graph theory to organize how many layers your chip must have to handle all of the interconnections, for example? Which properties of graphs come into play in such a circumstance?

**Week 4 DQ 3**

Trees occur in various venues in computer science: decision trees in algorithms, search trees, and so on. In linguistics, one encounters trees as well, typically as *parse trees*, which are essentially sentence diagrams, such as those you might have had to do in primary school, breaking a natural-language sentence into its components—clauses, subclauses, nouns, verbs, adverbs, adjectives, prepositions, and so on. What might be the significance of the depth and breadth of a parse tree relative to the sentence it represents? If you need to, look up *parse tree* and *natural language processing* on the Internet to see some examples

# Week 5 Individual Assignment Selected Textbook Exercises

**Complete **12 questions below.

· Ch. 15 of *Discrete and Combinatorial Mathematics*

o Supplementary Exercises, problems 1, 5, & 6

· Ch. 15 of *Discrete and Combinatorial Mathematics*

o Exercise 15.1, problems 1, 2, 11, 12, 14, & 15

· Ch. 15 of *Discrete and Combinatorial Mathematics*

- Exercise 15.1, problems 4, 5, 8, & 9

**Week 5 DQ 1**

How does Boolean algebra capture the essential properties of logic operations and set operations?

**Week 5 DQ 2**

How does the reduction of Boolean expressions to simpler forms resemble the traversal of a tree, from the Week Four material? What sort of Boolean expression would you end up with at the root of the tree?

**Week 5 DQ 3**

Conjunctive and disjunctive normal forms provide a form of balanced expression. How might this be important in terms of the efficiency of computational evaluation?

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