## Algebraic Expressions

For the following problems, write the number of terms that appear, then write the terms.

Exercise (PageIndex{1})

(4x^2 + 7x + 12)

**Answer**three: (4x^2, 7x, 12)

Exercise (PageIndex{2})

(14y^6)

Exercise (PageIndex{3})

(c + 8)

**Answer**two: (c, 8)

Exercise (PageIndex{4})

(8)

List, if any should appear, the common factors for the following problems.

Exercise (PageIndex{5})

(a^2 + 4a^2 + 6a^2)

**Answer**(a^2)

Exercise (PageIndex{6})

(9y^4 - 18y^4)

Exercise (PageIndex{7})

(12x^2y^3 + 36y^3)

**Answer**(12y^3)

Exercise (PageIndex{8})

(6(a+4) + 12(a+4))

Exercise (PageIndex{9})

(4(a+2b)+6(a+2b))

**Answer**(2(a+2b))

Exercise (PageIndex{10})

(17x^2y(z+4) + 51y(z+4))

Exercise (PageIndex{11})

(6a^2b^3c + 5x^2y)

**Answer**no common factors

For the following problems, answer the question of how many.

Exercise (PageIndex{12})

(x)'s in (9x)?

Exercise (PageIndex{13})

((a+b))'s in (12(a+b))?

**Answer**12

Exercise (PageIndex{14})

(a^4)'s in (6a^4)

Exercise (PageIndex{15})

(c^3)'s in (2a^2bc^3)?

**Answer**(2a^2b)

Exercise (PageIndex{16})

((2x+3y)^2)'s in (5(x+2y)(2x+3y)^3)?

For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors.

Exercise (PageIndex{17})

(8z, z)

**Answer**(8)

Exercise (PageIndex{18})

(16a^3b^2c^4, c^4)

Exercise (PageIndex{19})

(7y(y+3), 7y)

**Answer**((y+3))

Exercise (PageIndex{20})

((-5)a^5b^5c^5, bc)

## Equations

For the following problems, observe the equations and write the relationship being expressed.

Exercise (PageIndex{21})

(a = 3b)

**Answer**The value of (a) is equal to three times the value of (b).

Exercise (PageIndex{22})

(r = 4t + 11)

Exercise (PageIndex{23})

(f = dfrac{1}{2}m^2 + 6g)

**Answer**The value of (f) is equal to six times (g) more then one half times the value of (m) squared.

Exercise (PageIndex{24})

(x = 5y^3 + 2y + 6)

Exercise (PageIndex{25})

(P^2 = ka^3)

**Answer**The value of (P) squared is equal to the value of (a) cubed times (k).

Use numerical evaluation to evaluate the equations for the following problems.

Exercise (PageIndex{26})

(C = 2 pi r). Find (C) is (pi) is approximated by (3.14) and (r = 6)

Exercise (PageIndex{27})

(I = dfrac{E}{R}). Find (I) is (E = 20) and (R = 2).

**Answer**(10)

Exercise (PageIndex{28})

(I=prt). Find (I) if (p=1000), (r=0.06), and (t=3).

Exercise (PageIndex{29})

(E = mc^2). Find (E) if (m = 120) and (c = 186,000).

**Answer**(4.1515 imes 10^{12})

Exercise (PageIndex{30})

(z = dfrac{x-u}{s}). Find (z) if (x = 42), (u = 30), and (s = 12).

Exercise (PageIndex{31})

(R = dfrac{24C}{P(n+1)}). Find (R) if (C = 35), (P = 300), and (n = 19).

**Answer**(dfrac{7}{50}) or (0.14)

## Classification of Expressions and Equations

For the following problems, classify each of the polynomials as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term.

Exercise (PageIndex{32})

(2a+9)

Exercise (PageIndex{33})

(4y^3 + 3y + 1)

**Answer**trinomial, cubic; 4, 3, 1

Exercise (PageIndex{34})

(10a^4)

Exercise (PageIndex{35})

(147)

**Answer**monomial; zero; 147

Exercise (PageIndex{36})

(4xy + 2yz^2 + 6x)

Exercise (PageIndex{37})

(9ab^2c^2 + 10a^3b^2c^5)

**Answer**binomial; tenth; 9, 10

Exercise (PageIndex{38})

((2xy^3)^0, xy^3 ot = 0)

Exercise (PageIndex{39})

Why is the expression (dfrac{4x}{3x-7}) not a polynomial?

**Answer**... because there is a variable in the denominator

Exercise (PageIndex{40})

Why is the expression (5a^{dfrac{3}{4}}) not a polynomial?

For the following problems, classify each of the equations by degree. If the term linear, quadratic, or cubic applies, use it.

Exercise (PageIndex{41})

(3y + 2x = 1)

**Answer**linear

Exercise (PageIndex{42})

(4a^2 - 5a + 8 = 0)

Exercise (PageIndex{43})

(y - x - z + 4w = 21)

**Answer**linear

Exercise (PageIndex{44})

(5x^2 + 2x^2 - 3x + 1 = 19)

Exercise (PageIndex{45})

((6x^3)^0 + 5x^2 = 7)

**Answer**Quadratic

## Combining Polynomials Using Addition and Subtraction- Special Binomial Products

Simplify the algebraic expressions for the following problems.

Exercise (PageIndex{46})

(4a^2b + 8a^2b - a^2b)

Exercise (PageIndex{47})

(21x^2y^3 + 3xy + x^2y^3 + 6)

**Answer**(22x^2y^3 + 3xy + 6)

Exercise (PageIndex{48})

(7(x+1)+2x−6)

Exercise (PageIndex{49})

(2(3y^2+4y+4)+5y^2+3(10y+2))

**Answer**(11y^2 + 38y + 14)

Exercise (PageIndex{50})

(5[3x + 7(2x^2 + 3x + 2) + 5] - 10x^2 + 4(3x^2 + x))

Exercise (PageIndex{51})

(8{3[4y^3+y+2] + 6(y^3+2y^2)} - 24y^3 - 10y^2 - 3)

**Answer**(120y^3 + 86y^2 + 24y + 45)

Exercise (PageIndex{52})

(4a^2bc^3 + 5abc^3 + 9abc^3 + 7a^2bc^2)

Exercise (PageIndex{53})

(x(2x+5) + 3x^2 - 3x + 3)

**Answer**(5x^2 + 2x + 3)

Exercise (PageIndex{54})

(4k(3k^2 + 2k + 6) + k(5k^2 + k) + 16)

Exercise (PageIndex{55})

(2{5[6(b+2a+c^2)]})

**Answer**(60c^2 + 120a + 60b)

Exercise (PageIndex{56})

(9x^2y(3xy + 4x) - 7x^3y^2 - 30x^3y + 5y(x^3y + 2x))

Exercise (PageIndex{57})

(3m[5 + 2m(m+6m^2)] + m(m^2 + 4m + 1))

**Answer**(36m^4 + 7m^3 + 4m^2 + 16m)

Exercise (PageIndex{58})

(2r[4(r + 5) - 2r - 10] + 6r(r + 2))

Exercise (PageIndex{59})

(abc(3abc + c + b) + 6a(2bc + bc^2))

**Answer**(3a^2b^2c^2 + 7abc^2 + ab^2c + 12abc)

Exercise (PageIndex{60})

(s^{10}(2s^5 + 3s^4 + 4s^3 + 5s^2 + 2s + 2) - s^{15} + 2s^{14} + 3s(s^{12} + 4s^{11}) - s^{10})

Exercise (PageIndex{61})

(6a^4(a^2 + 5))

**Answer**(6a^6 + 30a^4)

Exercise (PageIndex{62})

(2x^2y^4(3x^2y + 4xy + 3y))

Exercise (PageIndex{63})

(5m^6(2m^7 + 3m^4 + m^2 + m + 1)

**Answer**(10m^{13} + 15m^{10} + 5m^8 + 5m^7 + 5m^6)

Exercise (PageIndex{64})

(a^3b^3c^4(4a + 2b + 3c + ab + ac + bc^2)

Exercise (PageIndex{65})

((x+2)(x+3))

**Answer**(x^2 + 5x + 6)

Exercise (PageIndex{66})

((y+4)(y+5))

Exercise (PageIndex{67})

((a+1)(a+3))

**Answer**(a^2 + 4a + 3)

Exercise (PageIndex{68})

((3x+4)(2x+6))

Exercise (PageIndex{69})

(4xy - 10xy)

**Answer**(-6xy)

Exercise (PageIndex{70})

(5ab^2 - 3(2ab^2 + 4))

Exercise (PageIndex{71})

(7x^4 - 15x^4)

**Answer**(-8x^4)

Exercise (PageIndex{72})

(5x^2 + 2x - 3 - 7x^2 - 3x - 4 - 2x^2 - 11)

Exercise (PageIndex{73})

(4(x-8))

**Answer**(4x-32)

Exercise (PageIndex{74})

(7x(x^2 - x + 3))

Exercise (PageIndex{75})

(-3a(5a - 6))

**Answer**(-15a^2 + 18a)

Exercise (PageIndex{76})

(4x^2y^2(2x-3y-5) - 16x^3y^2 - 3x^2y^3)

Exercise (PageIndex{77})

(-5y(y^2-3y-6) - 2y(3y^2+7) + (-2)(-5))

**Answer**(-11y^3 + 15y^2 + 16y + 10)

Exercise (PageIndex{78})

(-[-(-4)])

Exercise (PageIndex{79})

(−[−(−{−[−(5)]})])

**Answer**(-5)

Exercise (PageIndex{80})

(x^2 + 3x - 4 - 4x^2 - 5x - 9 + 2x^2 - 6)

Exercise (PageIndex{81})

(4a^2b - 3b^2 - 5b^2 - 8q^2b - 10a^2b - b^2)

**Answer**(-6a^2b - 8q^2b - 9b^2)

Exercise (PageIndex{82})

(2x^2 - x - (3x^2 - 4x - 5))

Exercise (PageIndex{83})

(3(a−1)−4(a+6))

**Answer**(-a - 27)

Exercise (PageIndex{84})

(−6(a+2)−7(a−4)+6(a−1))

Exercise (PageIndex{85})

Add (-3x + 4) to (5x - 8).

**Answer**(2x - 4)

Exercise (PageIndex{86})

Add (4(x^2 - 2x - 3)) to (-6(x^2 - 5)).

Exercise (PageIndex{87})

Subtract (3) times ((2x-1)) from (8) times ((x-4))

**Answer**(2x - 29)

Exercise (PageIndex{88})

((x+4)(x−6))

Exercise (PageIndex{89})

((x−3)(x−8))

**Answer**(x^2 - 11x + 24)

Exercise (PageIndex{90})

((2a−5)(5a−1))

Exercise (PageIndex{91})

((8b+2c)(2b−c))

**Answer**(16b^2 - 4bc - 2c^2)

Exercise (PageIndex{92})

((a-3)^2)

Exercise (PageIndex{93})

((3-a)^2)

**Answer**(a^2 - 6a + 9)

Exercise (PageIndex{94})

((x-y)^2)

Exercise (PageIndex{95})

((6x - 4)^2)

**Answer**(36x^2 - 48x + 16)

Exercise (PageIndex{96})

((3a-5b)^2)

Exercise (PageIndex{97})

((-x-y)^2)

**Answer**(x^2 + 2xy + y^2)

Exercise (PageIndex{98})

((k+6)(k−6))

Exercise (PageIndex{99})

((m+1)(m−1))

**Answer**(m^2 - 1)

Exercise (PageIndex{100})

((a−2)(a+2))

Exercise (PageIndex{101})

((3c+10)(3c−10))

**Answer**(9c^2 - 100)

Exercise (PageIndex{102})

((4a+3b)(4a−3b))

Exercise (PageIndex{103})

((5+2b)(5−2b))

**Answer**(25 - 4b^2)

Exercise (PageIndex{104})

((2y+5)(4y+5))

Exercise (PageIndex{105})

((y+3a)(2y+a))

**Answer**(2y^2 + 7ay + 3a^2)

Exercise (PageIndex{106})

((6+a)(6−3a))

Exercise (PageIndex{107})

((x^2 + 2)(x^2 - 3))

**Answer**(x^4 - x^2 - 6)

Exercise (PageIndex{108})

(6(a−3)(a+8))

Exercise (PageIndex{109})

(8(2y−4)(3y+8))

**Answer**(48y^2 + 32y - 256)

Exercise (PageIndex{110})

(x(x−7)(x+4))

Exercise (PageIndex{111})

(m^2n(m+n)(m+2n))

**Answer**(m^4n + 3m^3n^2 + 2m^2n^3)

Exercise (PageIndex{112})

((b+2)(b^2 - 2b + 3))

Exercise (PageIndex{113})

(3p(p^2 + 5p + 4)(p^2 + 2p + 7))

**Answer**(3p^5 + 21p^4 + 63p^3 + 129p^2 + 84p)

Exercise (PageIndex{114})

((a+6)^2)

Exercise (PageIndex{115})

((x-2)^2)

**Answer**(x^2 - 4x + 4)

Exercise (PageIndex{116})

((2x-3)^2)

Exercise (PageIndex{117})

((x^2 + y)^2)

**Answer**(x^4 + 2x^2y + y^2)

Exercise (PageIndex{118})

((2m - 5n)^2)

Exercise (PageIndex{119})

((3x^2y^3 - 4x^4y)^2)

**Answer**(9x^4y^6 - 24x^6y^4 + 16x^8y^2)

Exercise (PageIndex{120})

((a-2)^4)

## Terminology Associated with Equations

Find the domain of the equations for the following problems.

Exercise (PageIndex{121})

(y = 8x + 7)

**Answer**all real numbers

Exercise (PageIndex{122})

(y = 5x^2 - 2x + 6)

Exercise (PageIndex{123})

(y = dfrac{4}{x-2})

**Answer**all real numbers except 2

Exercise (PageIndex{124})

(m = dfrac{-2x}{h})

Exercise (PageIndex{125})

(z = dfrac{4x+5}{y+10})

**Answer**(x) can equal any real number; (y) can equal any number except (-10)

## Full-Length ALEKS Math Practice Test

Taking a Full-length ALEKS Math practice test is the best way to help you get familiar with the test format and feel more confident. Not only will this help you measure your exam readiness and solidify the concepts you’ve learned, but it is the best way to simulate test day.

To help you get the best out of this complete and realistic ALEKS Math practice test and prepare your mind and body for the actual test, we recommend that you treat this practice test as a real test. Prepare scratch papers, pencil, a timer, and a calculator and take the test in one sitting and follow the time limits to the minute.

Take the following full-length ALEKS Math practice test to simulate the test day experience. After you’ve finished, score your tests using the answer keys.**Good luck!**

1- Mr. Jones saves $2,500 out of his monthly family income of $55,000. What fractional part of his income does he save?

2- Four one – foot rulers can be split among how many users to leave each with (frac<1> <6>) of a ruler?

3- Simplify the expression.

((6x^3-8x^2+2x^4 )-(4x^2-2x^4+2x^3 ))

4- In two successive years, the population of a town is increased by (15\%) and (20\%). What percent of the population is increased after two years?

6- A cruise line ship left Port A and traveled 80 miles due west and then 150 miles due north. At this point, what is the shortest distance from the cruise to port A in miles? ____________

7- A shirt costing $200 is discounted (15\%). After a month, the shirt is discounted another (15\%). Which of the following expressions can be used to find the selling price of the shirt?

8- Solve: (5 + 8 imes (- 2) – [4 + 22 imes 5] div 6 = ? )

9- Which of the following points lies on the line (x+2y=4)?

10- 5 less than twice a positive integer is 83. What is the integer?

11- 11 yards 6 feet and 4 inches equals to how many inches?

12- Mr. Carlos family are choosing a menu for their reception. They have 3 choices of appetizers, 5 choices of entrees, 4 choices of cake. How many different menu combinations are possible for them to choose?

13- The average of five consecutive numbers is 38. What is the smallest number?

14- What is the difference of smallest 4–digit number and biggest 4–digit number?

15- How many tiles of 8 cm(^2 ) is needed to cover a floor of dimension 6 cm by 24 cm?

16- A ladder leans against a wall forming a (60^ circ ) angle between the ground and the ladder. If the bottom of the ladder is 30 feet away from the wall, how long is the ladder?

17- The average weight of 18 girls in a class is 60 kg and the average weight of 32 boys in the same class is 62 kg. What is the average weight of all the 50 students in that class?

18- An angle is equal to one fifth of its supplement. What is the measure of that angle?

19- In a stadium the ratio of home fans to visiting fans in a crowd is 5:7. Which of the following could be the total number of fans in the stadium?

20- If (40\%) of a class are girls, and (25\%) of girls play tennis, what percent of the class play tennis?

## Answers

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## 4.10: Exercise Supplement - Mathematics

div. A, title V, §591(b), Jan. 1, 2021, 134 Stat. 3665 , added item 2193b and struck out former item 2193b "Improvement of education in technical fields: program for support of elementary and secondary education in science, mathematics, and technology".

div. A, title V, §580(d)(3), Oct. 5, 1999, 113 Stat. 633 , added items 2192, 2193, 2193a, and 2193b and struck out former items 2192 "Science, mathematics, and engineering education" and 2193 "Science and mathematics education improvement program".

div. A, title II, §247(a)(2)(A), (C), Nov. 5, 1990, 104 Stat. 1523 , substituted "SUPPORT OF SCIENCE, MATHEMATICS, AND ENGINEERING EDUCATION" for "NATIONAL DEFENSE SCIENCE AND ENGINEERING GRADUATE FELLOWSHIPS" in chapter heading and added items 2192 to 2196.

#### Encouragement of Contractor Science, Technology, Engineering, and Mathematics (STEM) Programs

"(a) In General .&mdashThe Under Secretary of Defense for Research and Engineering, in coordination with the Under Secretary of Defense for Acquisition and Sustainment, shall develop programs and incentives to ensure that Department of Defense contractors take appropriate steps to&mdash

"(1) enhance undergraduate, graduate, and doctoral programs in science, technology, engineering, and mathematics (in this section referred to as &aposSTEM&apos)

"(2) make investments, such as programming and curriculum development, in STEM programs within elementary schools and secondary schools

"(3) encourage employees to volunteer in elementary schools and secondary schools, including schools that the Secretary of Defense determines serve high numbers or percentages of students from low-income families or that serve significant populations of military dependents, in order to enhance STEM education and programs

"(4) establish partnerships with appropriate entities, including institutions of higher education for the purpose of training students in technical disciplines

"(5) make personnel available to advise and assist in STEM educational activities aligned with functions of the Department of Defense

"(6) award scholarships and fellowships, and establish work-based learning programs in scientific disciplines

"(7) conduct recruitment activities to enhance the diversity of the STEM workforce or

"(8) make internships available to students of secondary schools, undergraduate, graduate, and doctoral programs in STEM disciplines.

"(b) Award Program .&mdashThe Secretary of Defense shall establish procedures to recognize defense industry contractors that demonstrate excellence in supporting STEM education, partnerships, programming, and other activities to enhance participation in STEM fields.

"(c) Implementation .&mdashNot later than 270 days after the date of the enactment of this Act [Jan. 1, 2021], the Under Secretary of Defense for Research and Engineering shall submit to the congressional defense committees [Committees on Armed Services and Appropriations of the Senate and House of Representatives] a report on the steps taken to implement the requirements of this section.

"(d) Definitions .&mdashIn this section:

"(1) The terms &aposelementary school&apos and &apossecondary school&apos have the meanings given those terms in section 8101 of the Elementary and Secondary Education Act of 1965 ( 20 U.S.C. 7801 ).

"(2) The term &aposinstitution of higher education&apos has the meaning given such term in section 101 of the Higher Education Act of 1965 ( 20 U.S.C. 1001 )."

div. A, title VIII, §862, Dec. 31, 2011, 125 Stat. 1521 , which related to the encouragement of contractor science, technology, engineering, and math (STEM) programs, was repealed by

CHAPTER 111 -SUPPORT OF SCIENCE, MATHEMATICS, AND ENGINEERING [email protected]!Sec. 2191 -->

### §2191. Graduate fellowships

(a) The Secretary of Defense shall prescribe regulations providing for the award of fellowships to citizens and nationals of the United States who agree to pursue graduate degrees in science, engineering, or other fields of study designated by the Secretary to be of priority interest to the Department of Defense.

(b) A fellowship awarded pursuant to regulations prescribed under subsection (a) shall be known as a "National Defense Science and Engineering Graduate Fellowship".

(c) National Defense Science and Engineering Graduate Fellowships shall be awarded solely on the basis of academic ability. The Secretary shall take all appropriate actions to encourage applications for such fellowships of persons who are members of groups (including minority groups, women, and disabled persons) which historically have been underrepresented in science and technology fields. Recipients shall be selected on the basis of a nationwide competition. The award of a fellowship under this section may not be predicated on the geographic region in which the recipient lives or the geographic region in which the recipient will pursue an advanced degree.

(d) The regulations prescribed under this section shall include&mdash

(1) the criteria for award of fellowships

(2) the procedures for selecting recipients

(3) the basis for determining the amount of a fellowship and

(4) the maximum amount that may be awarded to an individual during an academic year.

CHAPTER 111 -SUPPORT OF SCIENCE, MATHEMATICS, AND ENGINEERING [email protected]!Sec. 2192 -->

### §2192. Improvement of education in technical fields: general authority regarding education in science, mathematics, and engineering

(a) The Secretary of Defense, in consultation with the Secretary of Education, shall, on a continuing basis&mdash

(1) identify actions which the Department of Defense may take to improve education in the scientific, mathematics, and engineering skills necessary to meet the long-term national defense needs of the United States for personnel proficient in such skills and

(2) establish and conduct programs to carry out such actions.

(b)(1) In furtherance of the authority of the Secretary of Defense under any provision of this chapter or any other provision of law to support educational programs in science, mathematics, engineering, and technology, the Secretary of Defense may, unless otherwise specified in such provision&mdash

(A) enter into contracts and cooperative agreements with eligible entities

(B) make grants of financial assistance to eligible entities

(C) provide cash awards and other items to eligible entities

(D) accept voluntary services from eligible entities and

(E) support national competition judging, other educational event activities, and associated award ceremonies in connection with these educational programs.

(2) The Secretary of Defense may carry out the authority in paragraph (1) through the Secretaries of the military departments.

(A) The term "eligible entity" includes a department or agency of the Federal Government, a State, a political subdivision of a State, an individual, and a not-for-profit or other organization in the private sector.

(B) The term "State" means any State of the United States, the District of Columbia, the Commonwealth of Puerto Rico, Guam, the United States Virgin Islands, the Commonwealth of the Northern Mariana Islands, American Samoa, and any other territory or possession of the United States.

(c) The Secretary shall designate an individual within the Office of the Secretary of Defense to advise and assist the Secretary regarding matters relating to science, mathematics, and engineering education and training.

#### Amendments

added par. (2) and redesignated former par. (2) as (3).

added subsec. (b) and redesignated former subsec. (b) as (c).

amended section catchline generally. Prior to amendment, catchline read as follows: "Science, mathematics, and engineering education".

#### Science, Mathematics, and Research for Transformation (SMART) Defense Scholarship Pilot Program

div. A, title X, §1075(h)(5), Jan. 7, 2011, 124 Stat. 4377 , which related to a pilot program to provide financial assistance for education in science, mathematics, engineering, and technology skills and disciplines that were determined to be critical to the national security functions of the Department of Defense, was repealed and restated in section 2192a of this title by

#### Department of Defense Support for Science, Mathematics, and Engineering Education

div. A, title VIII, §829, Dec. 5, 1991, 105 Stat. 1444 , directed Secretary of Defense to develop and submit to Congress a master plan for activities by Department of Defense during each of fiscal years 1993 through 1997 to support education in science, mathematics, and engineering at all levels of education in the United States, with each such plan to be developed in consultation with Secretary of Education, prior to repeal by

CHAPTER 111 -SUPPORT OF SCIENCE, MATHEMATICS, AND ENGINEERING [email protected]!Sec. 2192a -->

### §2192a. Science, Mathematics, and Research for Transformation (SMART) Defense Education Program

(a) Requirement for Program .&mdashThe Secretary of Defense shall carry out a program to provide financial assistance for education in science, mathematics, engineering, and technology skills and disciplines that, as determined by the Secretary, are critical to the national security functions of the Department of Defense and are needed in the Department of Defense workforce.

(b) Financial Assistance .&mdash(1) Under the program under this section, the Secretary of Defense may award a scholarship or fellowship in accordance with this section to a person who&mdash

(A) is a citizen of the United States or, subject to subsection (g), a country the government of which is a party to The Technical Cooperation Program (TTCP) memorandum of understanding of October 24, 1995

(B) is pursuing an associates degree, undergraduate degree, or advanced degree in a critical skill or discipline described in subsection (a) at an accredited institution of higher education and

(C) enters into a service agreement with the Secretary of Defense as described in subsection (c).

(2) The amount of the financial assistance provided under a scholarship or fellowship awarded to a person under this subsection shall be an amount determined by the Secretary of Defense.

(3) Financial assistance provided under a scholarship or fellowship awarded under this section may be paid directly to the recipient of such scholarship or fellowship or to an administering entity for disbursement of the funds.

(c) Service Agreement for Recipients of Financial Assistance .&mdash(1) To receive financial assistance under this section&mdash

(A) in the case of an employee of the Department of Defense, the employee shall enter into a written agreement to continue in the employment of the department for the period of obligated service determined under paragraph (2) and

(B) in the case of a person not an employee of the Department of Defense, the person shall enter into a written agreement to accept and continue employment for the period of obligated service determined under paragraph (2)&mdash

(i) with the Department, including by serving on active duty in the Armed Forces or

(ii) with a public or private entity or organization outside of the Department if the Secretary&mdash

(I) is unable to find an appropriate position for the person within the Department and

(II) determines that employment of the person with such entity or organization for the purpose of such obligated service would provide a benefit to the Department.

(2) For the purposes of this subsection, the period of obligated service for a recipient of financial assistance under this section shall be the period determined by the Secretary of Defense as being appropriate to obtain adequate service in exchange for such financial assistance. The period of service required of a recipient may not be less than the total period of pursuit of a degree that is covered by such financial assistance. The period of obligated service is in addition to any other period for which the recipient is obligated to serve in the civil service of the United States.

(3) An agreement entered into under this subsection by a person pursuing an academic degree shall include any terms and conditions that the Secretary of Defense determines necessary to protect the interests of the United States or otherwise appropriate for carrying out this section.

(d) Employment of Program Participants .&mdashThe Secretary of Defense&mdash

(1) may, without regard to any provision of title 5 governing appointment of employees to competitive service positions within the Department of Defense, appoint to a position in the Department of Defense in the excepted service an individual who has successfully completed an academic program for which a scholarship or fellowship under this section was awarded and who, under the terms of the agreement for such scholarship or fellowship, at the time of such appointment, owes a service commitment to the Department

(2) may, upon satisfactory completion of 2 years of substantially continuous service by an incumbent who was appointed to an excepted service position under the authority of paragraph (1), convert the appointment of such individual, without competition, to a career or career conditional appointment and

(3) may establish arrangements so that participants may participate in a paid internship for an appropriate period with an industry sponsor.

(e) Refund for Period of Unserved Obligated Service .&mdash(1)(A) A participant in the program under this section who is not an employee of the Department of Defense and who voluntarily fails to complete the educational program for which financial assistance has been provided under this section, or fails to maintain satisfactory academic progress as determined in accordance with regulations prescribed by the Secretary of Defense, shall refund to the United States an appropriate amount, as determined by the Secretary.

(B) A participant in the program under this section who is an employee of the Department of Defense and who&mdash

(i) voluntarily fails to complete the educational program for which financial assistance has been provided, or fails to maintain satisfactory academic progress as determined in accordance with regulations prescribed by the Secretary or

(ii) before completion of the period of obligated service required of such participant&mdash

(I) voluntarily terminates such participant's employment with the Department or

(II) is removed from such participant's employment with the Department on the basis of misconduct,

shall refund the United States an appropriate amount, as determined by the Secretary.

(2) An obligation to reimburse the United States imposed under paragraph (1) is for all purposes a debt owed to the United States.

(3) The Secretary of Defense may waive, in whole or in part, a refund required under paragraph (1) if the Secretary determines that recovery would be against equity and good conscience or would be contrary to the best interests of the United States.

(4) A discharge in bankruptcy under title 11 that is entered less than five years after the termination of an agreement under this section does not discharge the person signing such agreement from a debt arising under such agreement or under this subsection.

(f) Relationship to Other Programs .&mdash(1) The Secretary of Defense shall coordinate the provision of financial assistance under the authority of this section with the provision of financial assistance under the other authorities provided in this chapter in order to maximize the benefits derived by the Department of Defense from the exercise of all such authorities.

(2) The Secretary of Defense shall seek to enter into partnerships with minority institutions of higher education and appropriate public and private sector organizations to diversify the participants in the program under subsection (a).

(g) Limitation on Participation .&mdash(1) The Secretary may not award scholarships or fellowships under this section to more than five individuals described in paragraph (2) per year.

(2) An individual described in this paragraph is an individual who&mdash

(A) has not previously been awarded a scholarship or fellowship under the program under this section

(B) is not a citizen of the United States and

(C) is a citizen of a country the government of which is a party to The Technical Cooperation Program (TTCP) memorandum of understanding of October 24, 1995.

You can't learn to speak a language by reading a textbook on grammar, and you can't learn to ride a bicycle by studying mechanics. In the same way, you can't learn algebra or any other mathematical subject just by reading a textbook and listening to lectures. You need to actively engage through the material, and one of the best ways to do that is with exercises.

Exercises will be posted each week, to be submitted *electronically* the following week. I will grade approximately four exercises each week from among those you submit (the actual number will depend on how much time the problems take to grade). Two of the problems I grade will be of your choice and the other two will be my choice. You should choose problems that are challenging but on which you have made meaningful progress.

I will grade your submissions based on the clarity of your writing, the correctness of your answers, and the appropriateness of the problems you have chosen.

Your grade on each assignment will be a single letter: A, B, C. These correspond roughly to `pass', `conditional pass', and `fail' on the prelims&mdashnot to the grades you will eventually receive in the course. I will supply comments, which I expect you to read and sometimes discuss with me (this is the whole point of my grading your exercises).

#### Guidelines for submitting assignments

- Assignments (after the first) must be typed using Latex and submitted electronically via D2L. You may submit the first electronically if you prefer.
**Do not submit homework by e-mail.** - Every assignment must include a section detailing references you have consulted. If you consult a reference and it contributes to what you write then you must cite it.
**Failure to do so is plagiarism!** - Indicate at the top of your submission the numbers of exactly two exercises you would like me to grade. If you don't do this, I will choose the two problems that seem easiest to grade.

#### Guidelines for collaboration and use of resources

I encourage you to collaborate! Your classmates will be your most valuable resource throughout graduate school. Talk to them, learn from them. But please follow the rules below when it comes to your homework assignments.

Likewise, it is good to look for other perspectives on the material we are studying. Learning math is a continuous process of reorganization in search of an intuitive perspective. Everyone is different and you may find that some texts speak to your intuition better than others.

That said, it is easy to find solutions to every exercise in Dummit and Foote online. Here is a link. It is easy to collaborate or to use online solutions in an unproductive or counterproductive way so while you *may* consult other resources, you must do so according to the rules below:

- Before discussing a problem or consulting an external resource, make an honest effort to solve it yourself.
- Make sure that when you talk to others or read outside material you are actively engaged in producing the solution and learning how it works. One way to do this is to try to improve the solution, either in the mathematical argument or its exposition.
- When you write up your solutions
**do not use any notes or other resources**other than your newly improved brain. This is the true test of whether you have improved your understanding. - You must list any discussions or resources that have contributed to your solutions in any way in the references section of your solutions.
**Failure to do this is plagiarism!** - Anything you submit as
*your*solution to an exercise must be reflection*your*understanding of that exercise and yours alone.

#### Week 1 (§1.1&mdash1.5): Due Monday, 8/31

§1.1: #5, 7, 20, 22, 24, 30 (see Example 6 on p. 18), 32

§1.2: supplement, #3 (try to do this without writing down the multiplication table), 7, 12

§1.3: #5, 7, 11, 15 (the hint is to use Exercise 10 of §3 and Exercise 24 of §1), 16

§1.4: #10

#### Week 2 (§1.6, 1.7, 2.1): Due Wednesday, 9/9

§1.6: #4, 6 (Hint: Look forward to §1.6, Ex. 21), 17, 23, supplement

§1.7: #8, 10a, 17, 18, 19, 21, 23

§2.1: #6 (Hint: consider the "infinite dihedral group"), 8, 10

#### Week 3 (§2.1&mdash2.4): Due Tuesday, 9/15

§1.3: supplement

§1.7: #13

§2.2: #7, 10, 14 (you can assume that F = &Ropf if you are unfamiliar with fields)

§2.3: #9, 12b, 18, 24, 25 (note: this map may not be a homomorphism!), 26

§2.4: #7, 12 (Writing out a multiplication table gets no credit for 7 and 12. Hint: use a presentation of the dihedral group.), 14cd, 15

#### Week 4 (§2.5, 3.1): Due Monday, 9/21

§2.5: #14 (just make the diagram no justification required you can hand in the diagram on paper if you want)

§3.1: #9, 12, 14, 19, 22, 25, 34, 35, 36, 42, supplement

#### Week 5 (Appendix I, §3.2&mdash3.4): Due Monday, 9/28

§2.4: #17

§3.2: #4, 9, 11, 14, 18, 22

§3.3: #4 (Hint: use universal properties!), 6, 7, 10

§3.4: #4

#### Week 6 (§3.4, 3.5, 4.1, 4.2): Due Monday, 10/5

§3.4: #5

§3.5: #10, 12

§4.1: #1, 9

§4.2: #7 (Hint: Every nontrivial subgroup of Q_{8} contains 〈-1〉), 8, 13 (Hint: compose the left regular permutation representation with the sign homomorphism and compute the image of an element of order 2)

§4.3: #17 (Hint: recognize D as the kernel of an action), 29

#### Week 7 (§4.3, 4.6): Due Monday, 10/12

§4.3: #13, 19, 25, 26 (Hint: use that if H is a subgroup of G and G is a union of conjugates of H then G = H), 30

§4.4: #14

§4.6: #4

#### Week 8 (§4.4, 4.5, 5.1, 5.2): Due Monday, 10/19

§4.4: #1, 20(a)

§4.5: #16, 19, 30, 32, 34

§5.1: #11

§5.2: #13 (observe that you are proving a universal property!)

#### Week 9 (§5.1&mdash5.5): Due Monday, 10/26

§4.4: #18

§5.2: #4bc, 14

§5.4: #5 (remember the universal property of the quotient by the commutator), 7 (this is good prelim practice), 11

§5.5: #11, 22, 23

#### Week 10 (§5.2, 6.3, review): Due Wednesday, 11/4

#### Week 11 (§7.1&mdash7.3): Due Monday, 11/9

Do up to 5 of these: §7.1, #3, 5, 6, 7, 15, 21 §7.2, #3b §7.3, #2, 11

Do these: §7.1, #10, 13bc §7.2, #2 §7.2, #3c, 5a, 7, 13 § 7.3, #12, 15

Do 5 less the number of problems you did from the first group: §7.1, #26, 27 §7.2, #5b §7.3, #14, 26c

#### Week 12 (§7.4&mdash7.6): Due Wednesday, 11/18

Do up to 5 of these: §7.3, #5 (remember all ring homomorphisms are unital), 7, 10, 22, 24 §7.4, #9, 15, 18 ~~§7.5, #7~~

Do these: §7.3, #29 §7.4, #2, 13 (for part (a), note that &phi -1 (P) = R is impossible when &phi is a unital homomorphism), 19, ~~39~~, **36** ~~§7.5, #6, 9~~

§7.5, #5

Do 5 less the number of problems you did from the first group: §7.3, #33 ~~§7.4, #33, 45, 46 (these three exercises requires a little topology)~~, **§7.4, #40** ~~§7.5, #11~~

#### Week 13 (§7.6, §8.1&mdash8.2, §9.2): Due Wednesday, 12/2

Do up to 3 of these: §7.6, #1, 7 §8.1, #7, 11b §8.2, #2, 6a supplement

Do these: §7.6, #6, supplement §8.1, #4 (for the first half of part (a), you do not need R to be a Euclidean domain), 6 (problem 4 may be helpful), 8

Do 3 less the number of problems you did from the first group: §7.6, #8, 10, 11, supplement

#### Week 14 (§9.4&mdash9.5): Due Monday, 12/14

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### Front matter

- Preface. Introductory note, copyright notice and main home page for all these pages.
- Contents. This page.
- Installation. Notes on installing or configuring a browser with MathML to view these pages.
- The contents pages of the published book. PDF taken from the published book, for people who haven't bought it yet!
- The preface from the published book. indicating the main aims of the book and how to read it.
- A (very short!) list of errata for the published book.
- The prfblock.sty LaTeX styles for typesetting proofs with the vertical lines as used in the book.

### Answers to selected exercises in the book

In due course I hope to have web pages containing answers or hints to all the exercises in the book. If the one you are looking for is not here yet please be patient.

### Additional exercises to test your knowledge and understanding

The following exercises test your knowledge and understanding of the marerial in the book further. They may be suitable for assessments for undergraduate courses, etc. Answers will not normally be provided on these web pages. They may not follow the order of the book exactly.

- Exercises and examples on simple formal systems. Systems similar to those in Chapter 3. You need to understand formal derivations and inductions on derivations.
- Exercises and examples on completeness and soundness for propositional logic. A complete baby propositional logic based on the arrow relation and the propositional constant bottom.
- Exercises and examples on the Sheffer stroke. A system for propositional logic based on a single propositional connective.
- Exercises and examples on first-order languages. Introductory exercises on first-order languages.

### Supplementary material on propositional and first order logic

This lists some supplementary material for propositional and first order logic, directly related to the book and available here as additional reading, some of it advanced . In particular this includes the proof of the Soundness Theorem, which is quite technical, especially when done properly. Just like the simpler examples of Soundness in the book, it is by induction on the length of proof. This sequence of web pages takes the reader through the material. Along the way you will find precise definitions of truth in an *L*-structure *M* and the precise definitions of substitution and the rules for first order logic. It is suitable for readers who want the full details and who have mastered most of *The Mathematics of Logic*.

- More on the König-lemma system. More on the 0,1-System of Chapter 3 of The Mathematics of Logic , including a way to avoid the use of Zorn's lemma and an alternative way of deriving König's lemma from the completeness theorem.
- Another 0-1 system. Some tricky variations on the 0-1 system to think about
- Between order and logic. Systems for lattices and theories of and and or , intermediate between the systems of chapters 4 and 6 of The Mathematics of Logic .
- Different propositional languages. Comparison of different languages and expressive completeness of propositional logical languages.
- Boolean terms and unique readability. The formal definition of the set of boolean terms over a set
*X*and the unique readability theorem for these. - Proofs as structured lists and proof trees. More on how to write a formal proof on a page.
- Free variables. The definition of free occurences of variables in a formula.
- Substitution in first-order language. The definition of valid substitution of terms for variables.
- Substitution and the rules for first order logic. The rules of first order logic revisited and made precise using the notion of substitution.
- The definition of truth. The definition of semantics for first order logic.
- Sematic aspects of substitution. Some preliminary results towards the soundness theorem.
- The proof of soundness. The proof of the Soundness Theorem for first order logic.

### The Gödel incompleteness theorems

Possibly the most celebrated results in logic, the incompleteness theorems show there are intrinsic limitations to the idea of mechanised proof. (In other words, mathematicians are not and never will be redundant!) The pages here sketch the details and the links with computability.

- Overview of the incompleteness theorems. Statements of the main theorems, and definitions of the key terms.
- Discretely ordered rings. A minimal algebraic theory of arithmetic.
- Exercises on discretely ordered rings.
- Exercises on discretely ordered rings - answers.
- Computability and the language of arithmetic. Connections between expressibility in the first order language of arithmetic and computability theory.
- Representability and diagonalisation. Representing functions and sets in a theory the Diagonalisation lemma.
- Gödel's first incompleteness theorem. The first application of Diagonalisation to incompleteness.
- The Gödel-Rosser incompleteness theorem. Rosser's trick to improve Gödel's first incompleteness theorem.
- Gödel's second incompleteness theorem. The non-provability of consistency.
- Interpretations. Interpretations of arithmetic in another theory, and the Gödel theorems for these theories.
- Hilbert's Programme.

### Axiomatic set theory

Axiomatic set theory is a first order theory into which all normal mathematics embeds. It formalises many arguments presented in *The Mathematics of Logic* including results on Zorn's Lemma and cardinal numbers.

- Introduction to axiomatic set theory. Basic axioms
- The cumulative hierachy of sets.
- The Axiom of Foundation.
- The Axiom of Infinity.
- The Axiom Scheme of Replacement with an application to transitive closure.
- Epsilon induction and recursion.
- Introduction to ordinals.
- Induction and recursion on ordinals.
- Ordinal arithmetic.
- The cumulative hierarchy and rank.
- The Axiom of Choice.
- Cardinals.
- König's inequality.
- Cofinality and inaccessibles.

### Some model theory

These pages build on Chapters 10 and 11 of The Mathematics of Logic. The goal is to give more examples and motivate the ideas of independence behind Morley's theorem.

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