Olympic Medals Consider the following table, which shows

The number of regions created when constructing a Venn diagram with three overlapping sets is .

a) When constructing a Venn diagram with three overlapping sets, region is generally completed first. b) When constructing a Venn diagram with three overlapping sets, after completing region V, the next regions generally completed are II, IV, and .

Complete DeMorgans laws: a) (A x B) = b) (A y B) =

In Exercises 1–4, fill in the blank with an appropriate word, phrase, or symbol(s). When using Venn diagrams to verify or determine whether set statements are equal we use _______________ reasoning.

A Venn diagram contains three sets, A, B, and C, as in Fig. 2.15 on page 68. If region V contains 4 elements and there are 12 elements in B y C, how many elements belong in region VI? Explain.

A Venn diagram contains three sets, A, B, and C, as in Fig. 2.15 on page 68. If region V contains 4 elements and there are 9 elements in A y B, how many elements belong in region II? Explain.

a) For U = 51, 2, 3, 4, 56, A = 51, 4, 56, and B = 51, 4, 56, does A x B = A y B? b) By observing the answer to part (a), can we conclude that A x B = A y B for all sets A and B? Explain. c) Using a Venn diagram, determine if A x B = A y B for all sets A and B.

Construct a Venn diagram illustrating the following sets. U = 5a, b, c, d, e, f, g, h, i, j6 A = 5c, d, e, g, h, i6 B = 5a, c, d, g6 C = 5c, f, i, j6

Construct a Venn diagram illustrating the following sets. U = 5Cinderella, Pinochio, Ratatouille, Fantasia, Dumbo, Bambi, Pocahontas, Hercules, Mulan, Tarzan, Cars6 A = 5Bambi, Hercules, Pocahontas, Tarzan6 B = 5Ratatouille, Bambi, Mulan, Hercules6 C = 5Pocahontas, Cinderella, Bambi, Ratatouille, Fantasia6

Construct a Venn diagram illustrating the following sets. U = 5 microwave oven, freezer, dishwasher, refrigerator, washer, dryer, toaster, blender, food processor, iron6 A = 5toaster, blender, iron, dishwasher, washer, dryer6 B = 5dishwasher, iron, freezer6 C = 5washer, dryer, iron, freezer, microwave oven6

Construct a Venn diagram illustrating the following sets. U = {American Eagle, Best Buy, Wal-Mart, Kmart, Target, Sears, JCPenney, Costco, Kohl’s, Gap, Gap Kids, Foot Locker, Old Navy, Macy’s} A = {American Eagle, Wal-Mart, Target, JCPenney, Old Navy} B = {Best Buy, Target, Costco, Old Navy, Macy’s} C = {Target, Sears, Kohl’s, Gap, JCPenney}

Construct a Venn diagram illustrating the following sets. U = 5 Louis Armstrong, Glenn Miller, Stan Kenton, Charlie Parker, Duke Ellington, Benny Goodman, Count Basie, John Coltrane, Dizzy Gillespie, Miles Davis, Thelonius Monk6 A = 5 Stan Kenton, Count Basie, Dizzy Gillespie, Duke Ellington, Thelonius Monk6 B = 5 Louis Armstrong, Glenn Miller, Count Basie, Duke Ellington, Miles Davis6 C = 5 Count Basie, Miles Davis, Stan Kenton, Charlie Parker, Duke Ellington6

Olympic Medals Consider the following table, which shows countries that won at least 25 medals in the 2008 Summer Olympics. Let the countries in the table represent the universal set. Let A = set of countries that won at least 30 gold medals. Let B = set of countries that won at least 15 silver medals. Let C = set of countries that won at least 10 bronze medals. Construct a Venn diagram that illustrates the sets A, B, and C.

Popular TV Shows Construct a Venn diagram illustrating the following sets. U = {American Idol (AI), CSI, Dancing with the Stars (DWS ), Family Guy (FG ), Gossip Girl (GG ), Monday Night Football (MNF ), NCIS, Sunday Night Football (SNF ), Survivor (S)}. A = {AI, CSI, DWS, SNF, NCIS} B = {AI, DWS, SNF, NCIS, MNF} C = {AI, CSI, SNF, NClS, MNF, S}

Rankings of Fruit-Producing Countries For Exercises 1520, use the following table, which shows the top 10 countries for production of apples, oranges, and nuts, The universal set is the set of countries listed in the table. Italy

Rankings of Fruit-Producing Countries For Exercises 15-20, use the following table, which shows the top 10 countries for production of apples, oranges, and nuts, The universal set is the set of countries listed in the table. Indicate in which region, I–VIII in Fig. 2.21, each of the following countries belongs. United States

Rankings of Fruit-Producing Countries For Exercises 1520, use the following table, which shows the top 10 countries for production of apples, oranges, and nuts, The universal set is the set of countries listed in the table. Canada

Rankings of Fruit-Producing Countries For Exercises 1520, use the following table, which shows the top 10 countries for production of apples, oranges, and nuts, The universal set is the set of countries listed in the table. Portugal

Rankings of Fruit-Producing Countries For Exercises 1520, use the following table, which shows the top 10 countries for production of apples, oranges, and nuts, The universal set is the set of countries listed in the table. Spain

Rankings of Fruit-Producing Countries For Exercises 1520, use the following table, which shows the top 10 countries for production of apples, oranges, and nuts, The universal set is the set of countries listed in the table. Mexico

Figures In Exercises 21-32, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 2132, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 2132, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 21–32, indicate in Fig. 2.22 the region in which each of the figures would be placed.

In Exercises 21–32, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 2132, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 2132, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 2132, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 2132, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 2132, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 2132, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Figures In Exercises 2132, indicate in Fig. 2.22 the region in which each of the figures would be placed.

Senate Bills During a session of the U.S. Senate, three bills were voted on. The votes of six senators are shown below the figure. Determine in which region of Fig. 2.23 each senator would be placed. The set labeled Bill 1 represents the set of senators who voted yes on Bill 1, and so on.Hutchinson yes no no

Senate Bills During a session of the U.S. Senate, three bills were voted on. The votes of six senators are shown below the figure. Determine in which region of Fig. 2.23 each senator would be placed. The set labeled Bill 1 represents the set of senators who voted yes on Bill 1, and so on.Kerry no no yes

Senate Bills During a session of the U.S. Senate, three bills were voted on. The votes of six senators are shown below the figure. Determine in which region of Fig. 2.23 each senator would be placed. The set labeled Bill 1 represents the set of senators who voted yes on Bill 1, and so on.McCain no no no

Senate Bills During a session of the U.S. Senate, three bills were voted on. The votes of six senators are shown below the figure. Determine in which region of Fig. 2.23 each senator would be placed. The set labeled Bill 1 represents the set of senators who voted yes on Bill 1, and so on.Mikulski yes yes yes

Senate Bills During a session of the U.S. Senate, three bills were voted on. The votes of six senators are shown below the figure. Determine in which region of Fig. 2.23 each senator would be placed. The set labeled Bill 1 represents the set of senators who voted yes on Bill 1, and so on. Rand no yes yes

Senate Bills During a session of the U.S. Senate, three bills were voted on. The votes of six senators are shown below the figure. Determine in which region of Fig. 2.23 each senator would be placed. The set labeled Bill 1 represents the set of senators who voted yes on Bill 1, and so on. Reid no yes no

In Exercises 39-52, use the Venn diagram in Fig. 2.24 to list the sets in roster form. A

In Exercises 39–52, use the Venn diagram in Fig. 2.24 to list the sets in roster form. U

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.26 B

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.27 C

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.28 A y B

In Exercises 39–52, use the Venn diagram in Fig. 2.24 to list the sets in roster form. \(A \cap C\)

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.30 (B y C) 4

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.31 A y B y C

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.32 A x B

In Exercises 39–52, use the Venn diagram in Fig. 2.24 to list the sets in roster form. \(B \cup C\)

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.34 (A x C)5

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.35 A y (B x C)

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.36 A

In Exercises 3952, use the Venn diagram in Fig. 2.24 to list the sets in roster form. Figure 2.37 (A x B x C)

In Exercises 5360, use Venn diagrams to determine whether the following statements are equal for all sets A and B.(A y B), Ax B

In Exercises 5360, use Venn diagrams to determine whether the following statements are equal for all sets A and B.(A y B), Ax B

In Exercises 5360, use Venn diagrams to determine whether the following statements are equal for all sets A and B. Ax B, A y B

In Exercises 5360, use Venn diagrams to determine whether the following statements are equal for all sets A and B.(A x B), (A y B)

In Exercises 53-60, use Venn diagrams to determine whether the following statements are equal for all sets A and B. \(A^{\prime} \cup B^{\prime},\) \((A \cup B)^{\prime}\)

In Exercises 5360, use Venn diagrams to determine whether the following statements are equal for all sets A and B. Ay B, A x B

In Exercises 5360, use Venn diagrams to determine whether the following statements are equal for all sets A and B. (Ay B), A x B

In Exercises 5360, use Venn diagrams to determine whether the following statements are equal for all sets A and B.Ay B, (Ay B)

In Exercises 6170, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C. A y (B x C), (A y B) x C

In Exercises 6170, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C.A x (B y C), (B y C) x A

In Exercises 6170, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C.A y (B x C), (B x C) y A

In Exercises 6170, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C.A x (B y C), Ay (Bx C)

In Exercises 6170, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C.A y (B x C), (A y B) x (A y C)

In Exercises 6170, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C.A x (B y C), (A x B) y (A x C)

In Exercises 6170, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C.A x (B x C), A x (By C)

In Exercises 6170, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C.(A x B) y (B x C), B x (A y C)

In Exercises 61-70, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C. \((A \cup B)^{\prime} \cap C\) \(\left(A^{\prime} \cup C^{\prime}\right) \cap\left(B^{\prime} \cup C\right)\)

In Exercises 6170, use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C. (C y B)x (A y B), A y (B y C)

In Exercises 7174, use set statements to write a description of the shaded area. Use union, intersection and complement as necessary. More than one answer may be possible.

In Exercises 7174, use set statements to write a description of the shaded area. Use union, intersection and complement as necessary. More than one answer may be possible.

In Exercises 7174, use set statements to write a description of the shaded area. Use union, intersection and complement as necessary. More than one answer may be possible.

In Exercises 7174, use set statements to write a description of the shaded area. Use union, intersection and complement as necessary. More than one answer may be possible.

Let U = 51, 2, 3, 4, 5, 6, 7, 8, 9, 106 A = 51, 2, 3, 46 B = 53, 6, 76 C = 56, 7, 96 a) Show that (A x B) y C = (A y C) x (B y C) for these sets. b) Make up your own sets A, B, and C. Verify that (A x B) y C = (A y C) x (B y C) for your sets A, B, and C. c) Use Venn diagrams to verify that (A x B) y C = (A y C) x (B y C) for all sets A, B, and C.

Let U = 5a, b, c, d, e, f, g, h, i6 A = 5a, c, d, e, f6 B = 5c, d6 C = 5a, b, c, d, e6 a) Determine whether (A x C)y B = (A y C)y B for these sets. b) Make up your own sets, A, B, and C. Determine whether (A x C)y B = (A y C)y B for your sets. c) Determine whether (A x C)y B = (A y C)y B for all sets A, B, and C. 77.

Blood Types A hematology text gives the following information on percentages of the different types of blood worldwide. Construct a Venn diagram similar to the one in Example 2 and place the correct percentage in each of the eight regions.

Define each of the eight regions in Fig. 2.25 using sets A, B, and C and a set operation. (Hint: A y By C defines region I.) Figure 2.25 U A B I II III IV VI V VII C VIII

Categorizing Contracts J & C Mechanical Contractors wants to classify its projects. The contractors categorize set A as construction projects, set B as plumbing projects, and set C as projects with a budget greater than $300,000. a) Draw a Venn diagram that can be used to categorize the company projects according to the listed criteria. b) Determine the region of the diagram that contains construction projects and plumbing projects with a budget greater than $300,000. Describe the region using sets A, B, and C with set operations. Use union, intersection, and complement as necessary. c) Determine the region of the diagram that contains plumbing projects with a budget greater than $300,000 that are not construction projects. Describe the region using sets A, B and C with set operations. Use union, intersection, and complement as necessary. d) Determine the region of the diagram that contains construction projects and nonplumbing projects whose budget is less than or equal to $300,000. Describe the region using sets A, B, and C with set operations. Use union, intersection, and complement as necessary.

We were able to determine the number of elements in the union of two sets with the formula n(A x B) = n(A) + n(B) - n(A y B). Can you determine a formula for finding the number of elements in the union of three sets? In other words, write a formula to determine n(A x B x C). [Hint: The formula will contain each of the following: n(A), n(B), n(C), n(A y B y C), n(A y By C), n(Ay B y C), and 2n(A y B y C). R

a) Construct a Venn diagram illustrating four sets, A, B, C, and D. (Hint: Four circles cannot be used, and you should end up with 16 distinct regions.) Have fun! b) Label each region with a set statement (see Exercise 78). Check all 16 regions to make sure that each is distinct.

The two Venn diagrams below illustrate what happens when colors are added or subtracted. Do research in an art text, an encyclopedia, the Internet, or another source and write a report explaining the creation of the colors in the Venn diagrams, using such terms as union of colors and subtraction (or difference) of colors. Magenta Yellow Blue Subtractive color mixing Red Green Additive color mixing Blueviolet