1. A set 𝐶 is defined to be set difference between two set 𝐴 and 𝐵: 𝐶 = 𝐴 − 𝐵. And set 𝐴 and 𝐵 are defined as like below. 𝐴: 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝐵: 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑐𝑖 is an element of the set 𝐶. Check the statements below by each and verify whether they are TRUE or FALSE, and shortly explain why. a. For any element 𝑐𝑖 ∈ 𝐶, it holds 𝑐𝑖 2 > 0 b. There is an element 𝑐𝑗 ∈ 𝐶, satisfying this condition; 𝑐𝑗 𝑖𝑠 𝑎 𝑟𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 c. 𝜋 ∈ 𝐶 d. One of the two solutions of equation (𝑥 − 2) (𝑥 + 7 9 ) = 0 is an element of the set 𝐶 e. 𝐶 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑒𝑡 f. 𝐵⋂𝐶 = ∅ g. For any two different elements 𝑐𝑖 and 𝑐𝑗 in the set 𝐶, their product is also an element of 𝐶: 𝑐𝑖𝑐𝑗 ∈ 𝐶 (𝑐𝑖 ≠ 𝑐𝑗) 2

1

 

Problem set 1

 

Due date: February 13th Tuesday 14:00 (before beginning of the class)

 

You can write your answers by hand or in a word (or Latex) file. You can submit your

 

assignment in person (written or printed), before beginning of class on the Tuesday. Or, you

 

can upload your submission on Canvas. When you upload your assignment, please check

 

resolution of your file. And I recommend you to upload it as .pdf file.

 

You can cooperate with others or can utilize some materials on the internet. But you have

 

to submit your own work individually. You need to clarify process of your work, especially

 

for calculation.

 

1. A set 𝐶 is defined to be set difference between two set 𝐴 and 𝐵: 𝐶 = 𝐴 − 𝐵. And set 𝐴 and

 

𝐵 are defined as like below.

 

𝐴: 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝐵: 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠

 

𝑐𝑖 is an element of the set 𝐶. Check the statements below by each and verify whether they are

 

TRUE or FALSE, and shortly explain why.

 

a. For any element 𝑐𝑖 ∈ 𝐶, it holds 𝑐𝑖 2 > 0

 

b. There is an element 𝑐𝑗 ∈ 𝐶, satisfying this condition; 𝑐𝑗 𝑖𝑠 𝑎 𝑟𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔 𝑑𝑒𝑐𝑖𝑚𝑎𝑙

 

c. 𝜋 ∈ 𝐶

 

d. One of the two solutions of equation (𝑥 − 2) (𝑥 + 7

 

9 ) = 0 is an element of the set 𝐶

 

e. 𝐶 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑒𝑡

 

f. 𝐵⋂𝐶 = ∅

 

g. For any two different elements 𝑐𝑖 and 𝑐𝑗 in the set 𝐶, their product is also an element of 𝐶:

 

𝑐𝑖𝑐𝑗 ∈ 𝐶 (𝑐𝑖 ≠ 𝑐𝑗)

 

2

 

2. Textbook exercise 2.3.4. from (a) to (g)

 

3. Textbook exercise 2.4.6.

 

4. Check the statements below by each and verify whether they are TRUE or FALSE, and

 

shortly explain why.

 

a. A relation 𝑥2 + 𝑦2 = 9 (𝑥| − 3 ≤ 𝑥 ≤ 3) is a function.

 

b. There is a function 𝑦 = 3𝑥 + 1 and domain of 𝑥 is all integers. Then, image of 𝑦 includes

 

some irrational numbers.

 

c. 𝑥3 ÷ 𝑥7 = 𝑥10

 

d. 𝑥√2(𝑥2√2 + 𝑥√3) = 𝑥4+√6

 

e. 𝑥𝑎 × 𝑥𝑏 × 𝑥𝑐 = 𝑥𝑎𝑏𝑐

 

f. 𝑎𝑥 × 𝑏𝑥 × 𝑐𝑥 = (𝑎𝑏𝑐)𝑥

 

5. Find equilibrium market price and quantity the market given below

 

𝑄𝑑 = 𝑄𝑠 𝑄𝑑 = 30 − 2𝑃 𝑄𝑠 = −6 + 4𝑃

 

6. Textbook exercise 3.2.2.(b)

 

7. Textbook exercise 3.4.3.

 

8. Find solution of system of equations. If solution is not unique, then shortly explain why.

 

a. 𝑥 − 4𝑦 = −3 2𝑥 + 3𝑦 = 7 b. 3𝑥 + 2𝑦 = 5 4𝑥 + 8

 

3 𝑦 = 6

 

3

 

9. Answer below matrix operations. If it cannot be defined, shortly explain why.

 

a. [ 1 2 3 4

 

] + [ −1 −2 5 0

 

] b. 3 [ 0 −3 3 0

 

] − 2 [ −1 −2 1 7

 

]

 

c. [ 1 2 3 4

 

] [ −2 3 0 5 −8 1

 

] d. [ 5 0 1 2 1 −3

 

] [ 3 0 2

 

−3 2 8 ]

 

e. [ −2 3 0 5 −8 1

 

] [ 1 2 3 4

 

]

 

10. Textbook exercise 4.2.6.

 

11. Check linear dependence of three vectors.

 

𝑢 = (−2, 3), 𝑣 = (5, 1), 𝑤 = (1, −2)

 

12. For two vectors 𝑢 = (3,2) and 𝑣 = (−1, 7), find scalars 𝑎, 𝑏, 𝑐, and 𝑑 satisfying below

 

equations. If you cannot find those scalars, then shortly explain why.

 

a. 𝑎𝑢 + 𝑏𝑣 = (1,0) b. 𝑐𝑢 + 𝑑𝑣 = (0,1)

 

13. Prove (𝐴′)−1 = (𝐴−1)′. You can prove this statement by utilizing matrix multiplication,

 

and properties of transpose and inverse. You do not have to introduce dimension of matrix 𝐴.

 

14. Find determinant of below matrices

 

a. [7] b. [ 1 2 3 4

 

]

 

c. [ 1 0 2

 

−1 5 5 −3 0 7

 

] d. [ 1 2 3 4 5 6 0 0 0

 

] e. [ 1 2 7

 

−6 5 0 −6 2 2

 

]

 

4

 

15. Find rank of below matrices

 

a. [ 6 8 3

 

4 1] b. [

 

1 2 5 1 3 9 2 0 4

 

] c. [ 1 0 0 0 1 0 0 0 1

 

]

 

16. Check the statements below by each and verify whether they are TRUE or FALSE, and

 

shortly explain why.

 

a. If 𝐴 is a square matrix, then both of 𝐴𝐴′ and 𝐴′𝐴 are well defined.

 

b. If 𝐴 is a (2 × 2) nonsingular matrix, then its rank is equal to 2.

 

c. If determinant of (3 × 3) matrix 𝐴 is zero, then its rank cannot be 3.

 

d. For any (3 × 3) symmetric matrix 𝐴, 𝐴2 is also a symmetric matrix.

 

e. If 𝐴𝐵 is well defined, then 𝐴′𝐵′ is also well defined.

 

f. For a (2 × 2) matrix 𝐴, if 𝐴2 = 𝐼 then 𝐴 = 𝐼

 

17. Textbook exercise 5.4.2. only (a) and (b)

 

18. Textbook exercise 5.4.5. only (a) and (b)

 

19. Find 𝑥2 ∗ by applying Cramer’s rule. If you cannot find a unique value of 𝑥2

 

∗, then shortly

 

explain why.

 

a.

 

3𝑥1 + 2𝑥2 + 4𝑥3 = 2

 

2𝑥1 − 4𝑥2 + 𝑥3 = 3

 

2𝑥1 − 1𝑥2+ = −4

 

b.

 

𝑥1 + 2𝑥2 + 3𝑥3 = 1

 

5𝑥1 − 1𝑥2 + 𝑥3 = −3

 

2𝑥1 − 1𝑥2 + 3𝑥3 = 2