addition of matrices

proof

Using simple properties of integers and of l.c.m. and h.c.f. we can easily show that axioms (1)-(3) given in the definition of a Boolean algebra are satisfied. Now axiom (4) will hold if and only if for any a ϵ B, a and n/a have no common factor, other than 1. This condition is equivalent to n being square-free.

Let B be the set of all positive integers which are divisors of

Let B be the set of all positive integers which are divisors of 70; i.e., B = {1, 2, 5, 7, 10, 14, 35, 70}. For any a, b ϵ B, let a + b = l.c.m of a, b; a ∙ b = h.c.f. of a, b and a’ = ⁷<span style=’font-size: 50%’>/₀. Then with the help of elementary properties of l.c.m. and h.c.f. it can be easily verified that (B, +, ∙, ‘, 1, 70) is a Boolean algebra. Here 1 is the zero element and 70 is the unit element.

Let A be a non-empty set and P(A) be the power set of A.

Let A be a non-empty set and P(A) be the power set of A. Then P(A) is a Boolean algebra under the usual operations of union, intersection and complementary in P(A). The sets ∅ and A are the zero element and unit element of the Boolean algebra P(A). Observe that if A is an infinite set, then the Boolean algebra P(A) will contain infinite number of elements.

FREQUENCY DISTRIBUTION

FREQUENCY DISTRIBUTION A grouping of quantitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

How do we develop a frequency distribution? The following example shows the steps to construct a frequency distribution. Remember, our goal is to construct tables, charts, and graphs that will quickly summarize the data by showing the location, extreme values, and shape of the data’s distribution.

TABLE 2–4 Profit on Vehicles Sold Last Month by the Applewood Auto Group Maximum

Minimum

$1,387 $2,148 $2,201 $ 963 $ 820 $2,230 $3,043 $2,584 $2,370 1,754 2,207 996 1,298 1,266 2,341 1,059 2,666 2,637 1,817 2,252 2,813 1,410 1,741 3,292 1,674 2,991 1,426 1,040 1,428 323 1,553 1,772 1,108 1,807 934 2,944 1,273 1,889 352 1,648 1,932 1,295 2,056 2,063 2,147 1,529 1,166 482 2,071 2,350 1,344 2,236 2,083 1,973 3,082 1,320 1,144 2,116 2,422 1,906 2,928 2,856 2,502 1,951 2,265 1,485 1,500 2,446 1,952 1,269 2,989 783 2,692 1,323 1,509 1,549 369 2,070 1,717 910 1,538 1,206 1,760 1,638 2,348 978 2,454 1,797 1,536 2,339 1,342 1,919 1,961 2,498 1,238 1,606 1,955 1,957 2,700 443 2,357 2,127 294 1,818 1,680 2,199 2,240 2,222 754 2,866 2,430 1,115 1,824 1,827 2,482 2,695 2,597 1,621 732 1,704 1,124 1,907 1,915 2,701 1,325 2,742 870 1,464 1,876 1,532 1,938 2,084 3,210 2,250 1,837 1,174 1,626 2,010 1,688 1,940 2,639 377 2,279 2,842 1,412 1,762 2,165 1,822 2,197 842 1,220 2,626 2,434 1,809 1,915 2,231 1,897 2,646 1,963 1,401 1,501 1,640 2,415 2,119 2,389 2,445 1,461 2,059 2,175 1,752 1,821 1,546 1,766 335 2,886 1,731 2,338 1,118 2,058 2,487

S O L U T I O N

To begin, we need the profits for each of the 180 vehicle sales listed in Table 2–4. This information is called raw or ungrouped data because it is simply a listing

E X A M P L E

Ms. Kathryn Ball of the Applewood Auto Group wants to summarize the quantitative variable profit with a frequency distribution and display the distribution with charts and graphs. With this information, Ms. Ball can easily answer the following ques- tions: What is the typical profit on each sale? What is the largest or maximum profit on any sale? What is the smallest or minimum profit on any sale? Around what value do the profits tend to cluster?

DESCRIBING DATA

DESCRIBING DATA: FREQUENCY TABLES, FREQUENCY DISTRIBUTIONS, AND GRAPHIC PRESENTATION 25

(a) Is the data qualitative or quantitative? Why? (b) What is the table called? What does it show? (c) Develop a bar chart to depict the information. (d) Develop a pie chart using the relative frequencies.

The answers to the odd-numbered exercises are at the end of the book in Appendix D.

1. A pie chart shows the relative market share of cola products. The “slice” for Pepsi- Cola has a central angle of 90 degrees. What is its market share?

2. In a marketing study, 100 consumers were asked to select the best digital music player from the iPod, the iRiver, and the Magic Star MP3. To summarize the con- sumer responses with a frequency table, how many classes would the frequency table have?

3. A total of 1,000 residents in Minnesota were asked which season they preferred. One hundred liked winter best, 300 liked spring, 400 liked summer, and 200 liked fall. Develop a frequency table and a relative frequency table to summarize this information.

4. Two thousand frequent business travelers are asked which midwestern city they prefer: Indianapolis, Saint Louis, Chicago, or Milwaukee. One hundred liked India- napolis best, 450 liked Saint Louis, 1,300 liked Chicago, and the remainder pre- ferred Milwaukee. Develop a frequency table and a relative frequency table to summarize this information.

5. Wellstone Inc. produces and markets replacement covers for cell phones in five different colors: bright white, metallic black, magnetic lime, tangerine orange, and fusion red. To estimate the demand for each color, the company set up a kiosk in the Mall of America for several hours and asked randomly selected people which cover color was their favorite. The results follow:

E X E R C I S E S

Bright white 130 Metallic black 104 Magnetic lime 325 Tangerine orange 455 Fusion red 286

a. What is the table called? b. Draw a bar chart for the table. c. Draw a pie chart. d. If Wellstone Inc. plans to produce 1 million cell phone covers, how many of

each color should it produce? 6. A small business consultant is investigating the performance of several companies.

The fourth-quarter sales for last year (in thousands of dollars) for the selected com- panies were:

Fourth-Quarter Sales Company ($ thousands)

Hoden Building Products $ 1,645.2 J & R Printing Inc. 4,757.0 Long Bay Concrete Construction 8,913.0 Mancell Electric and Plumbing 627.1 Maxwell Heating and Air Conditioning 24,612.0 Mizelle Roofing & Sheet Metals 191.9

The consultant wants to include a chart in his report comparing the sales of the six companies. Use a bar chart to compare the fourth-quarter sales of these corpora- tions and write a brief report summarizing the bar chart.

DESCRIBING DATA: FREQUENCY TABLES, FREQUENCY DISTRIBUTIONS, AND GRAPHIC PRESENTATION 23

DESCRIBING DATA: FREQUENCY TABLES, FREQUENCY DISTRIBUTIONS, AND GRAPHIC PRESENTATION 23

Pie and bar charts both serve to illustrate frequency and relative frequency ta- bles. When is a pie chart preferred to a bar chart? In most cases, pie charts are used to show and compare the relative differences in the percentage of observations for each value or class of a qualitative variable. Bar charts are preferred when the goal is to compare the number or frequency of observations for each value or class of a qualitative variable. The following Example/Solution shows another application of bar and pie charts.

E X A M P L E

SkiLodges.com is test marketing its new website and is interested in how easy its website design is to navigate. It randomly selected 200 regular Internet users and asked them to perform a search task on the website. Each person was asked to rate the relative ease of navigation as poor, good, excellent, or awesome. The re- sults are shown in the following table:

Awesome 102 Excellent 58 Good 30 Poor 10

1. What type of measurement scale is used for ease of navigation? 2. Draw a bar chart for the survey results. 3. Draw a pie chart for the survey results.

A set of multiple choice intermediate algebra questions,

A set of multiple choice intermediate algebra questions,

      1. If f(x) = 4x3 – 4x2 + 10, then f(-2) =
        A. 26 B. -38 C. 10 D. 38

        2. Which of these values of x satisfies the inequality -7x + 6 ≤ -8

        A. -2 B. 0 C. -7 D. 2

    1. The domain of the function f(x) = √(6 – 2x) is given by
      A. x > 0 x ≥ 6 C. x ≤ 3 D. x ≤ 6

    2. The lines y = 2x and 2y = – x are
      A. parallel B. perpendicular
      C. horizontal D. vertical

    3. The equation |-2x – 5| – 3 = k has no solution if k =
      A. -5 B. -3 C. 7 D. 0

    4. The inequality corresponding to the statement:“the price is no less than 100 Dollars” is
      A. x < 100 B. x ≥ 100 C. x ≤ 100 D. x > 100

    1. Which of these relations DOES NOT represent a function?
      A. {(2,3),(-4,3),(7,3)} B. {(0,0),(-1,-1),(2,2)}
      C. {(2,3),(-5,3),(2,7)} D. {(-1,3),(-5,3),(-9,0)}

    2. Which of these points DOES NOT lie on the graph of y = -x + 3?
      A. (9,6) B. (3,0) C. (-2,5) D. (2,2)

    3. What is the slope of the line perpendicular to the line y = -5x + 9?
      A. 5 B. -5 C. 1/5 D. -1/5

    4. Which property is used to write:3(xy) = (3x)y?
      A. Commutative property of multiplication B. Multiplicative inverse property
      C. Distributive property D. Associative property of multiplication

    5. In which quadrant do the lines x = 3 and y = -4 intersect?
      A. I B. II C. III D. IV

    6. The value of 2-|-2| is
      A. 4 B. 0.25 C. -4 D. -0.25

  1. If a and b are positive real numbers, then (a0 – 3b0)5 =
    A. 0 B. 1 C. -32 D. 32

  2. Which inequality describes the situation:“length L is at most 45 cm”.
    A. L = 45 cm B. L > 45 cm C. L ≥ 45 cm D. L ≤ 45 cm

  3. The equation mx – 8 = 6 – 7(x + 3) DOES NOT have any solution if m =
    A. 3 B. 7 C. -7 D. 0

  4. The equation – mx + 1 = 13 – 4(x + 3) is an identity if m =
    A. 4 B. -4 C. 1 D. -1

  5. Which of the following is ALWAYS true?
    A. A function is not a relation B. Every function is a relation
    C. Every relation is a function D. A relation is not a function

  6. Which of these inequalities has NO solutions?

  7. The lines y = (a – 5)x + 5 and y = -2x + 7 are parallel if a =
    A. -2 B. 3 C. 5 D. -5

  8. The lines y = (a – 5)x + 5 and y = -2x + 7 are perpendicular if a =
    A. 11/2 B. 5 C. -2/9 D. 9/2

Intermediate Algebra Questions

Intermediate Algebra Questions

    1. Write 1.5 × 10-5 in standard form.
    2. Evaluate: 30 – |-x + 6| for x = 10
    1. Evaluate: 2xy3 + x – 2y for x = 2 and y = -2
    2. What is the slope of the line perpendicular to the line y = – 4
    3. Write an equation of the line with slope 2 and x-intercept (-4 , 0).
    4. Solve the equation: -3(-x + 5) + 20 = -10(x – 3) + 4
    5. Solve the inequality: 4(x – 6) + 4 < 8(x – 4)
    6. Solve the equation: 3(x – 2)2 – 12 = 0
    7. Solve the equation: x / 3 + 2 / 7 = x / 7 – 5
    8. Line L is defined as line through the point (2 , 7) and perpendicular to the line x + y = 0. What is the point of intersection of L and the line x + y = 0
    9. What is the point of intersection of the lines: x + 2y = 4 and -x – 3y = -7?
    10. How many solutions do the system of equations 2x – 3y = 4 and 4x – 6y = -7 have?
    11. For what value(s) of A does the system of equation A x + 6y = 0 and 2x – 7y = 3 have no solutions?
    12. Solve |2x – 4| – 2 = 6.
    13. How many solutions does the equation 2x2 + 3x = 8 have?
    14. Solve the equation 3x2 + 6x – 1 = 8.
    15. Solve the system of equations: 2x + 5y = 18 and -3x – y = -1.
    16. What is the range of function f defined by: f = {(2,3),(1,4),(5,4),(0,3)}
    17. Factor the expression 2x2 + 3x + 1.
    18. Factor the expression 10x2 + 20x – 80.

True/False Algebra Questions

True/False Algebra Questions

    1. (True or False)     The inequality |x + 1| < 0 has no solution.
    2. (True or False)     If a and b are negative numbers, and |a| < |b|, then b – a is negative.
    3. (True or False)     The equation 2x + 7 = 2(x + 5) has one solution.
    4. (True or False)     The multiplicative inverse of -1/4 is -1/8.
    5. (True or False)     x ÷ (2 + z) = x ÷ 2 + x ÷ z
    1. (True or False)     |-8| – |10| = -18
    2. (True or False)     (8 ÷ 4) ÷ 2 = 8 ÷ (4 ÷ 2)
    3. (True or False)     31.5(1.004)20 < 31.6(1.003)25
    4. (True or False)     The graph of the equation y = 4 has no x-intercept.
    5. (True or False)     The value of n(n + 3)/2 = 3/2 when n = 0.
    6. (True or False)     The distance between the numbers -9 and 20 is equal to the distance between 9 and -20 on the number line.
    7. (True or False)     If f(x) = sqrt(1 – x), then f(-3) = 2.
    8. (True or False)     The slope of the line 2x + 2y = 2 is equal to 2.
    9. (True or False)     |x + 5| is always positive.
    10. (True or False)     The distance between the points (0 , 0) and (5 , 0) in a rectangular system of axes is 5.
    11. (True or False)     1 / (2x – 4) is undefined when x = -4.
    12. (True or False)     (-1/5)-2 = 25.
    13. (True or False)     The reciprocal of 0 is equal to 0.
    14. (True or False)     The additive inverse of -10 is equal to 10.
    15. (True or False)     1 / (x – 4) = 1/x – 1/4.