Consider a set A = { 1, 2, 3, 4, 5, 6, 8}

Consider these relations,

1. R1 = { ( a, b) | a = 3b }

Can you write down the pairs ? one such pair is ( 3, 1)

2. R2 = { (a, b) | 2a = b }

Can you write down the pairs ? one such pair is (2, 4)

3. R3 = { ( a, b) | a >= 2b }

Can you write down the pairs ? one such pair is (4, 2)

4. R4 = { (a, b) | a + b < 4 }

Can you write down the pairs ?

Consider this set A = { a, b, c, d } and the following relations

R6 = { ( a, a ), ( a, b), ( b, b), ( c, d ) }

R7 = { ( a, a), ( b, b ), ( b, c ), ( c, c ), ( c, d), (d, d) }

R8 = { (a, b), (a, d), ( b, a), ( d, a) , ( b, d) , (d, b) }

R9 = { ( a, a), ( b, c) }

R10 = { ( a, b), (b, d), (a, d), ( a, a ) , (b, b), (b, d) }

5. Which of the above relations are reflexive and state why ?  

6. Which of the above relations are symmetric and state why ?

7. Which of the above relations are transitive and state why ?

A relation is equivalence when the relation is : reflexive, symmetric AND Transitive.

Please NOTE it is AND. It has to be all three.

8. Is below mentioned R12 a Equivalence relation ? and state why or why not ?  

Consider this set S = { 1, 2 , 3}

R12 = { ( 1, 1), ( 1, 2 ), (2, 1) , (2 , 2 ), (3, 3), (1, 3) , (3, 1) }

1. A pigment solution contains compound A with 4 polar groups and compound B with 2 polar groups. You plan to separate these compounds using paper chromatography with a nonpolar solvent. Predict the location of the two bands relative to the solvent front. Explain your answer.

2. Paper chromatography is used to separate pigments extracted in acetone from fresh spinach. Explain how differences in polarity are used to separate the pigments in this chromatography experiment.

Program P1

1) integer A, B;

2) input (A);

3) while (A > 0)

4) {

5) B = 1;

6) if (A < 10)

7) B = 0;

8) if (A < 20 or A > 25)

9) B = A * B;

10) else

11) B = A + B;

12) output (A, B);

13) input (A);

14) }

15) output (“Program ends.”);

16) end;

T = {t1=<1>, t2=<33>, t3=<‐1>} or T = {t1=, t2=, t3=}

1. What is the statement domain for P1? Express as line numbers. Exclude syntactical markers, such as {, }, else, and end.

2. What is the statement coverage of T for P1? Express as an unsimplified fraction.

1. Design a FA for the language L = {w : na(w) ≤ 4}.

2. Design a FA that accepts all strings that contain the substring aba.

3. Design a FA for the language L = {aⁱ ba^j : i, j ≥ 0, i + j is odd}

4. Design a FA for the language L = {bⁿa^m : n ≤ 2, m ≥ 3}

Let A = {1, 2, 3, 4, 5}. Find the inverse of the following functions f: A→ A.

1. 1. ? = {(1,1),(2,3),(3,2),(4,4),(5,5)    
2. 2. ? = {(1,5),(2,4),(3,2),(4,1),(5, 4)}
3. 3. ? = {(2,1),(3,4),(1,3),(4,1),(5, 2)}

The 4 most fundamental factors that affect the cost of money are

(1) production opportunities, (2) time preferences for consumption, (3) risk, and (4) opportunity cost.

a. True b. False

On January 1, 2018, NRC Credit Corporation leased equipment to Brand Services under a finance/sales-type lease designed to earn NRC a 13% rate of return for providing long-term financing. The lease agreement specified:

Ten annual payments of $68,000 beginning January 1, 2018, the beginning of the lease and each December 31 thereafter through 2026.

The estimated useful life of the leased equipment is 10 years with no residual value. Its cost to NRC was $380,163.

The lease qualifies as a finance lease/sales-type lease.

A 10-year service agreement with Quality Maintenance Company was negotiated to provide maintenance of the equipment as required. Payments of $6,000 per year are specified, beginning January 1, 2018. NRC was to pay this cost as incurred, but lease payments reflect this expenditure.

A partial amortization schedule, appropriate for both the lessee and lessor, follows:

(FV of $1, PV of $1, FVA of $1, PVA of $1, FVAD of $1 and PVAD of $1) (Use appropriate factor(s) from the tables provided.)

Required:
1. Prepare the appropriate entries for the lessee related to the lease on January 1, 2018 and December 31, 2018.
2. Prepare the appropriate entries for the lessor related to the lease on January 1, 2018 and December 31, 2018.

On December 31, 2019, Sumner Company held Wall Company bonds in its portfolio of trading securities. The bonds have a par value of $40,000, carry a 10% annual interest rate, mature in 2026, and had originally been purchased at par. The market value of the bonds on December 31, 2019 was $38,000.

On January 1, 2020, Sumner acquired bonds of Doherty Company with a par value of $30,000 for $30,200. The Doherty Company bonds carry an annual interest rate of 12% and mature on December 31, 2024. Additionally, on the same date, Sumner acquired Maggio Company bonds with a face value of $20,000 for $19,500. The Maggio Company bonds carry an 8% annual interest rate and mature on December 31, 2029. At the end of 2020, the respective market values of the bonds were: Wall, $39,000; Doherty, $31,000; and Maggio, $21,000. Sumner classifies all of the debt securities as trading. Assume that Sumner uses the straight-line method to amortize any discounts or premiums.

Required:

2 fair four-sided dice with faces numbered 1, 2, 3, 4 are rolled. Random variable D is the nonnegative difference |X2 −X1| of the numbers rolled on the 2 dice and Y is a Bernoulli random variable such that Y = 0 if the sum X1 +X2 is even and and Y = 1 if the sum is odd.

a. Find joint probability mass function of D and Y.

b. Are D and Y dependent or independent? How do you know?