StudyDaddy Article Writing

Answered You can hire a professional tutor to get the answer.

QUESTION

Apr 11, 2020

Hello, I am looking for someone to write an article on Week 1. It needs to be at least 500 words.

Hello, I am looking for someone to write an article on Week 1. It needs to be at least 500 words. Individual Selected Textbook Exercises Directions Use the e book to see examples, ebook user is salvatore2 & password is salvatore80 Ch. 1of Discrete and Combinatorial Mathematics

Supplementary Exercises

1. In the manufacture of a certain type of automobile, four

kinds of major defects and seven kinds of minor defects can

occur. For those situations in which defects do occur, in how

many ways can there be twice as many minor defects as there

are major ones?

Answer: For the ratio of 1:2 to occur, there are a total of 3 situations as follows:

a. 1 major defect : 2 minor defect

For 1 major defect: 4!/(4-1)! = 4

For 2 minor defect: 7!/(7-2)! = 7x6 =42

Hence: 4 x 42 = 168

b. 2 major defect : 4 minor defect

For 2 major defect: 4!/(4-2)! = 4 x 3 = 12

For 4 minor defect: 7!/(7-4)! = 7x6x5x4 =840

Hence: 12 x 840 = 10,080

c. 3 major defect : 6 minor defect

For 3 major defect: 4!/(4-3)! = 4 x 3 x 2 = 24

For 6 minor defect: 7!/(7-6)! = 7x6x5x4x3x2 =5040

Hence: 24 x 5040 = 120,960

Totally: 168 + 10,080 + 120,960 = 131,208

2. A machine has nine different dials, each with five settings

labeled 0, 1, 2, 3, and 4.

a) In how many ways can all the dials on the machine be

set?

Answer: 59 = 1,953,125

b) If the nine dials are arranged in a line at the top of the

machine, how many of the machine settings have no two

adjacent dials with the same setting?

Answer: 1st place = 5 possibilities

2nd,3rd to 9th = 4 possibilities each place

Hence: 5 x 48 = 327,680

3. How many of the 9000 four-digit integers 1000, 1001,

1002, . . . , 9998, 9999 have four distinct digits that are either

increasing (as in 1347 and 6789) or decreasing (as in

6421 and 8653)?

Answer:

For increasing, note that there are important points to be considered such as the digit 9 can only be found in the last place, the digit 0 is unusable hence our effective digits is nine in total (1,2,3,4,5,6,7,8 and 9) and that order and distinction is very important. In this method used, the tens place is used as reference:

A. Digit in the tens place

B. Combinations

Note that digit 1 is not possible to put in the tens place as the resulting number &lt. 1000 which is the leftmost boundary.

To proceed, we note how many digits can be placed in the units place considering the digit in the tens place. Here we are restricted to the digits 1,2,3,4,5 and 6 since the digit 7,8,9 will compromise the whole restriction of increasing order when use it in the units position:

A. Digit in the tens place

C. Possible Number of Digits in Units Place

D. Total combination = (B*C)

Total

=126

For decreasing, note that there are important points to be considered such as the digit 9 can only be placed in the units position and that the digit 0 can be used but only placed in the thousands position. In this method used, the tens place is used as reference:

E. Digit in the tens place

F. Combinations

Note that digit 1 is not possible to put in the tens place as the resulting number would violate the restriction of decreasing and distinct numbers.

B. Digit in the tens place

G. Possible Number of Digits in Units Place

H. Total combination = (B*C)

Total

=210

4. Mr. and Mrs. Richardson want to name their new daughter

so that her initials (first, middle, and last) will be in alphabetical

order with no repeated initial. How many such triples of initials

can occur under these circumstances?

Answer:

Grouping the alphabets by 3 in succession such as ABC,BCD,EFG and limiting it with three distinct initials and alphabetical in order means that we can end only on the combination XYZ. Hence, on the 26 alphabets, only 24 combinations can be used given the restriction.

Ch. 2 of Discrete and Combinatorial Mathematics

5. Let p, q be primitive statements for which the implication

p→q is false. Determine the truth values for each of the following.

a) p ∧ qp=1, q=0. hence truth value =0 (false)

b) ￢p ∨ q p=0, q=0. hence truth value =0 (false)

c) q →p q=0, p=1. hence truth value =1 (true)

d) ￢q →￢p q=1, p=0. hence truth value =0 (false)

6. Use the substitution rules to verify that each of the following

is a tautology. (Here p, q, and r are primitive statements.)

a) [p ∨ (q ∧ r)] ∨ ￢[p ∨ (q ∧ r)]

Let m = [p ∨ (q ∧ r)], hence: m ∨ ￢m

To illustrate

￢m

m ∨ ￢m

Hence: [p ∨ (q ∧ r)] ∨ ￢[p ∨ (q ∧ r)] is a tautology since all components have truth values=1

b) [(p ∨ q)→r] ↔ [￢r →￢(p ∨ q)]

Let s = (p ∨ q) hence: [s→r] ↔ [￢r →￢s]

s→r

￢r

￢s

￢r →￢s

Since truth values for components of s→r corresponds to ￢r →￢s, then [s→r] ↔ [￢r →￢s] is a tautology.

7. For any statements p, q, prove that

a) ￢(p ↓ q)⇐⇒(￢p ↑￢q)

using Tautology:

(p ↓ q)

￢(p ↓ q)

￢p

￢q

(￢p ↑￢q)

Since the values correspond and the conditions of tautology exists, then

￢(p ↓ q)⇐⇒(￢p ↑￢q) which is De Morgan’s Law

b) ￢(p ↑ q)⇐⇒(￢p ↓￢q)

using Tautology:

(p ↑ q)

￢(p ↑ q)

￢p

￢q

(￢p ↓￢q)

Since the values correspond and the conditions of tautology exists, then

￢(p ↑ q)⇐⇒(￢p ↓￢q)

8. The following are three valid arguments. Establish the validity

of each by means of a truth table. In each case, determine

which rows of the table are crucial for assessing the validity of

the argument and which rows can be ignored.

a) [p ∧ (p→q) ∧ r] → [(p ∨ q)→r]

(p→q)

p ∧ (p→q)

p ∧ (p→q) ∧ r

p ∨ q

(p ∨ q)→r

[p ∧ (p→q) ∧ r] → [(p ∨ q)→r]

Green shade : necessary

Blue shade : unnecessary

Since the indicator → indicates that the truth value can only be false (=0) when the statement on the left side is true, then there was no need to investigate further the rows shaded blue.

b) [[(p ∧ q)→r] ∧ ￢q ∧ (p→￢r)]→(￢p ∨￢q)

(p ∧ q)

(p ∧ q)→r

￢p

￢q

￢r

[(p ∧ q)→r] ∧ ￢q

p→￢r

[[(p ∧ q)→r] ∧ ￢q ∧ (p→￢r)]

￢p ∨￢q

[[(p ∧ q)→r] ∧ ￢q ∧ (p→￢r)]→(￢p ∨￢q)

Green shade : necessary

Blue shade : unnecessary

Since the indicator → indicates that the truth value can only be false (=0) when the statement on the left side [[(p ∧ q)→r] ∧ ￢q ∧ (p→￢r)] is true (1), there was no need to investigate those with values =0.

c) [[p ∨ (q ∨ r)] ∧ ￢q]→(p ∨ r)

(q ∨ r)

[p ∨ (q ∨ r)]

￢q

[[p ∨ (q ∨ r)] ∧ ￢q]

(p ∨ r)

[[p ∨ (q ∨ r)] ∧ ￢q]→(p ∨ r)

Green shade : necessary

Blue shade : unnecessary

Since the indicator → indicates that the truth value can only be false (=0) when the statement on the left side [[p ∨ (q ∨ r)] ∧ ￢q] is true (1), there was no need to investigate those with values =0.

Ch. 3 of Discrete and Combinatorial Mathematics

9. a) Determine the sets A, B where A − B _ {1, 3, 7, 11},

B − A _ {2, 6, 8}, and A ∩ B _ {4, 9}.

From A ∩ B € {4, 9}, we have a preliminary A € {4, 9} and B € {4, 9}

We add to set A the values of A − B € {1, 3, 7, 11} then A € {1, 3,4, 7, 9,11}

We add to set B the values of B − A € {2, 6, 8} then B € {2,4,6,8,9}

b) Determine the sets C, D where C − D €{1, 2, 4},

D − C € {7, 8}, and C ∪ D € {1, 2, 4, 5, 7, 8, 9}

From C − D €{1, 2, 4}, we have a preliminary C €{1, 2, 4}

From D − C € {7, 8}, we have a preliminary D € {7, 8}

Considering C ∪ D € {1, 2, 4, 5, 7, 8, 9} to be the entirety of the sets, then

Answer:

the set C must be {1, 2, 4, 5, 9}

the set D must be {5,7,8, 9}

10. UsingVenn diagrams, investigate the truth or falsity of each of the following, for sets A, B, C ⊆ _.

a) A (B ∩ C) = (A B) ∩ (A C)

Venn Diagram for A (B ∩ C): Venn Diagram for (A B) ∩ (A C):

Hence the statement is false.

b) A − (B ∪ C) = (A − B) ∩ (A − C)

Venn Diagram for A − (B ∪ C): Venn Diagram for (A − B) ∩ (A − C):

From the diagram, we can see that the statement is true.

c) A (B C) = (A B) C

Venn Diagram for A (B C): Venn Diagram for (A B) C:

From the diagram, we can see that the statement is true.

11. During freshman orientation at a small liberal arts college, two showings of the latest James Bond movie were presented. Among the 600 freshmen, 80 attended the first showing and 125

attended the second showing, while 450 didn’t make it to either showing. How many of the 600 freshmen attended twice?

The possible Venn Diagram is shown in the following:

The intersection represents the number of students that attended twice. Hence, the answer is that 55 students attended twice.

12. A manufacturer of 2000 automobile batteries is concerned about defective terminals and defective plates. If 1920 of her batteries have neither defect, 60 have defective plates, and 20

have both defects, how many batteries have defective terminals?

The possible Venn Diagram is shown in the following:

From the resulting Venn Diagram, one can verify that 60 automobiles have defective plates and 20 have both defects thus satisfying the given. 2000 – 1920 – 60 = 20 which is the number of automobiles which has defective terminals only. Therefore, the number of automobiles with defective batteries is 20 + 20 = 40.

Answer is 40.