Discrete Math

1. Prove each of the following for all n > by the Principal of Mathematical Induction.

1. . \[{{1}^{2}}+{{3}^{2}}+{{5}^{2}}...+{{(2n-1)}^{2}}=\frac{n(2n-1)(2n+1)}{3}\]
2. . \(1*3+2*4+3*5+...+n(n+2)=\frac{n(n+1)(2n+7)}{6}\)

2. Establish each of the following for all n > 1 by the Principal of Mathematical Induction.

b). \(\sum\limits_{i=1}^{n}{i({{2}^{i}})=2+(n-1){{2}^{n+1}}}\)

4.2

1. The integer sequence \({{a}_{1,}}{{a}_{2,}}{{a}_{3,}}...\) defined explicitly by the formula \({{a}_{n}}={{5}_{n}}\) for n \(\in {{\mathbb{Z}}^{+}}\), can also be defined recursively by

1. \({{a}_{1}}=5;and\)
2. \({{a}_{n+1}}={{a}_{n}}+5,\) for n > 1.

For the integer sequence \({{b}_{1,}}{{b}_{2,}}{{b}_{3,}}...\) where \({{b}_{n}}=n(n+2)\) for all n \(\in {{\mathbb{Z}}^{+}}\), we can also provide the recursive definition:

1. ’ \(b=3;an{{d}_{{}}}\)
2. ’ \({{b}_{n+1}}={{b}_{n}}+2n+3,\) for n > 1,

Give a recursive definition for each of the following integer sequences \({{c}_{1,}}{{c}_{2,}}{{c}_{3}}...\) where for all n \(\in {{\mathbb{Z}}^{+}}\) we have

c) \({{c}_{n}}=3n+7\)

d) \({{c}_{n}}=7\)

e) \({{c}_{n}}={{n}^{2}}\)

f) \({{c}_{n}}=2-{{(-1)}^{n}}\)

8a). Develop a recursive definition for the addition of n real numbers \({{x}_{1,}}{{x}_{2,}}....,{{x}_{n}}\), where n > 2.

16. Give a recursive definition for the set of all

1. positive even integers
2. nonnegative even integers

5.1

2. If A = {1,2,3}, and B = {2,4,5}, give examples of

b) three nonempty relations in A

4. For which sets A, B is it true that A x B = B x A?

6. The men’s final at Wimbledon is won by the first player to win three sets of the five-set match. Let C and M denote the players. Draw a tree diagram to show all the ways in which the match can be decided.

8. Logic chips are taken from a container, tested individually, and labeled defective or good. The testing process is continued until either two defective chips are found or five chips are tested in total. Using a tree diagram, exhibit a sample space for this process.

10. A rumor is spread as follows. The originator calls two people. Each of these people phones three friends, each of whom in turn calls five associates. If no one receives more than one call, and no one calls the originator, how many people now know the rumor? How many phone calls were made?

12. Let A, B be sets with |B| = 3. If there are 4096 relations from A to B, what is |A|?

5.2

2. Does the formula \(f(x)=1/({{x}^{2}}-2)\) define a function \(f:R\to R\) ? A function \(\mathbb{Z}\to R\) ?

4. If there are 2187 functions \(f:A\to B\) and |B| = 3, what is |A|?

6. Let A, B, C \(\subseteq {{\mathbb{Z}}^{2}}\) where A ={(x,y)|y = 2x+1}, B = {(x,y)|y = 3x}, and C = {(x,y)|x-y = 7}.

1. Determine

1. \(A\cap C\)
2. \(\overline{B\cap C}\)
3. \(\bar{A}\cup \bar{C}\)
4. \(\bar{B}\cup \bar{C}\)

8. Determine whether each of the following statements is true or false. If the statement is false, provide a counterexample.

1. . \(_{-}{{a}_{-}}{{=}^{-}}{{a}^{-}}\) for all \(a\in \mathbb{Z}\)

c) \(_{-}{{a}_{-}}{{=}^{-}}{{a}^{-}}\) -1 for all \(a\in R-\mathbb{Z}\)

16. Let \(f:R\to R\) where \(f(x)={{x}^{2}}\). Determine f(A) for the following subsets A taken from the domain R.

1. A = {2,3}

c) A = {-3,3}

e) A = {-7,2}

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