Set Theory Concepts and Venn Diagrams Alıştırma Soruları — GRE General Test — Sayfa 2

Soru 21Soru →

A survey of $500$ university researchers evaluated their usage of three high-performance computing resources: Cloud Containers ( $C$ ), GPU Accelerators ( $G$ ), and Distributed Storage ( $D$ ). Exactly $50$ researchers use none of these three resources. The survey revealed that equal numbers of researchers use Cloud Containers and GPU Accelerators ( $|C| = |G| = 240$ ), while $210$ researchers use Distributed Storage ( $|D| = 210$ ). Exactly $40$ researchers use all three resources. Furthermore, the number of researchers who use both $C$ and $G$ but not $D$ is equal to the number who use both $G$ and $D$ but not $C$ , and this quantity is exactly twice the number of researchers who use both $C$ and $D$ but not $G$ .

How many researchers use GPU Accelerators ( $G$ ) ONLY?

32

72

104

112

144

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

72 researchers use GPU Accelerators (

G

) only.

The total number of researchers using at least one resource is

500 - 50 = 450

. Assigning

x

to the region using Cloud Containers and Distributed Storage only, the regions for C-and-G-only and G-and-D-only are each

2x

. Applying the three-set inclusion-exclusion principle gives

450 = 240 + 240 + 210 - (5x + 120) + 40

, which simplifies to

5x = 160

, so

x = 32

. The exclusive overlapping regions containing GPU Accelerators are

2(32) = 64

and

2(32) = 64

. Subtracting these overlaps along with the triple intersection (

40

) from

|G| = 240

yields

240 - (64 + 64 + 40) = 72

.

Adım Adım Çözüm

1

Determine the total number of researchers using at least one resource.

|C \cup G \cup D| = 500 - 50 = 450

.

Subtract researchers who use none of the resources from the total surveyed.

2

Define variables for the two-set exclusive intersection regions.

Let

|C \cap D \cap G^c| = x

. Then

|C \cap G \cap D^c| = 2x

and

|G \cap D \cap C^c| = 2x

.

The problem states that the C-and-D-only region is half of the other two exclusive two-set intersection regions.

3

Express the full pairwise intersections including the triple intersection (

|C \cap G \cap D| = 40

).

|C \cap G| = 2x + 40

,

|G \cap D| = 2x + 40

, and

|C \cap D| = x + 40

.

Each pairwise intersection is the sum of its exclusive two-set intersection and the three-set intersection.

4

Apply the 3-Set Principle of Inclusion-Exclusion.

450 = 240 + 240 + 210 - [(2x + 40) + (2x + 40) + (x + 40)] + 40 \implies 450 = 690 - (5x + 120) + 40 \implies 450 = 610 - 5x \implies 5x = 160 \implies x = 32

.

Inclusion-exclusion formula:

|C \cup G \cup D| = |C| + |G| + |D| - (|C \cap G| + |G \cap D| + |C \cap D|) + |C \cap G \cap D|

.

5

Calculate the number of researchers using GPU Accelerators (

G

) only.

Exclusive

G

-only

= |G| - (|C \cap G \cap D^c| + |G \cap D \cap C^c| + |C \cap G \cap D|) = 240 - (64 + 64 + 40) = 240 - 168 = 72

.

Subtract all overlapping regions within set

G

from the total size of set

G

.

Anahtar Kavram

3-Set Inclusion-Exclusion Principle and Venn Diagram Region Partitioning

Soru 22Soru →

A market research firm surveyed a group of $200$ consumers regarding their active subscriptions to three digital services: FilmStream ( $F$ ), AudioVibe ( $A$ ), and PrintPlus ( $P$ ). Every surveyed consumer subscribed to at least one of the three services. The survey gathered the following information:
- $120$ consumers subscribed to FilmStream
- $100$ consumers subscribed to AudioVibe
- $75$ consumers subscribed to PrintPlus
- $45$ consumers subscribed to both FilmStream and AudioVibe
- $35$ consumers subscribed to both AudioVibe and PrintPlus
- $30$ consumers subscribed to both FilmStream and PrintPlus

How many consumers subscribed to all three digital services?

15

25

35

55

95

Cevabı ve açıklamayı göster

Cevap: $15$

Cevap

15 consumers subscribed to all three digital services.

According to the 3-set inclusion-exclusion principle,

|F \cup A \cup P| = |F| + |A| + |P| - (|F \cap A| + |A \cap P| + |F \cap P|) + |F \cap A \cap P|

. Substituting the given values gives

200 = 120 + 100 + 75 - (45 + 35 + 30) + |F \cap A \cap P|

, which simplifies to

200 = 185 + |F \cap A \cap P|

. Subtracting

185

from

200

yields

15

.

Adım Adım Çözüm

1

Apply the Principle of Inclusion-Exclusion for three sets.

|F \cup A \cup P| = |F| + |A| + |P| - (|F \cap A| + |A \cap P| + |F \cap P|) + |F \cap A \cap P|

This formula accounts for elements counted multiple times across overlapping sets.

2

Substitute the known values into the equation.

200 = 120 + 100 + 75 - (45 + 35 + 30) + |F \cap A \cap P|

Every consumer subscribes to at least one service, so

|F \cup A \cup P| = 200

.

3

Simplify the numerical terms on the right side of the equation.

200 = 295 - 110 + |F \cap A \cap P| \implies 200 = 185 + |F \cap A \cap P|

Sum of individual sets is

295

, and sum of pairwise intersections is

110

.

4

Solve for the target three-set intersection

|F \cap A \cap P|

.

|F \cap A \cap P| = 200 - 185 = 15

Isolating the variable gives the number of consumers in all three sets.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets

Tahmini Süre:1m 30s

Soru 23Soru →

A biotechnology consortium surveyed $180$ research laboratories regarding their implementation of three diagnostic platforms: Platform $A$ , Platform $B$ , and Platform $C$ .

- $95$ laboratories use Platform $A$ .
- $85$ laboratories use Platform $B$ .
- $80$ laboratories use Platform $C$ .
- $15$ laboratories use all three platforms.
- $20$ laboratories use none of the three platforms.
- The number of laboratories that use Platform $A$ and Platform $B$ but NOT Platform $C$ is equal to the number of laboratories that use Platform $B$ and Platform $C$ but NOT Platform $A$ .
- $30$ laboratories use Platform $A$ and Platform $C$ but NOT Platform $B$ .

How many laboratories use Platform $A$ ONLY?

15

20

30

45

50

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30 laboratories use Platform A only.

Subtracting the 20 laboratories that use none of the platforms from the total population of 180 gives a union size of 160. Applying the inclusion-exclusion formula yields

160 = (95 + 85 + 80) - (x + 15 + x + 15 + 30 + 15) + 15

, which resolves to

x = 20

laboratories that use Platforms A and B only. Subtracting all overlap regions from Platform A's total gives

95 - 20 - 30 - 15 = 30

laboratories that use Platform A only.

Adım Adım Çözüm

1

Calculate the total number of laboratories that use at least one platform.

|A \cup B \cup C| = 180 - 20 = 160

Laboratories using at least one platform equal the total surveyed minus those using none.

2

Set up the Principle of Inclusion-Exclusion for three sets.

|A \cup B \cup C| = |A| + |B| + |C| - (|A \cap B| + |B \cap C| + |A \cap C|) + |A \cap B \cap C|

Standard formula relating total union size to individual set sizes and intersections.

3

Express pairwise intersections in terms of non-overlapping regions and solve for the unknown region

x

.

Let

x

be the number of labs using Platform

A

and

B

only. Given

A \cap C \text{ only} = 30

and

A \cap B \cap C = 15

, we have:

160 = (95 + 85 + 80) - [(x + 15) + (x + 15) + (30 + 15)] + 15

160 = 260 - (2x + 75) + 15 = 200 - 2x \implies 2x = 40 \implies x = 20

Substituting the given equality of regions allows finding

x = 20

.

4

Calculate the number of laboratories that use Platform

A

only.

\text{Platform } A \text{ only} = |A| - (A \cap B \text{ only}) - (A \cap C \text{ only}) - (A \cap B \cap C) = 95 - 20 - 30 - 15 = 30

Subtracting all overlap regions containing Platform A from the total Platform A count gives the single-region count.

Anahtar Kavram

Three-Set Principle of Inclusion-Exclusion and Venn Diagram Region Partitioning

Soru 24Soru →

An executive education program surveyed $400$ professional candidates regarding their enrollment in three specialized tracks: Artificial Intelligence ( $A$ ), Financial Technology ( $F$ ), and Sustainable Energy ( $S$ ).

The survey revealed the following data:
- $40$ candidates were enrolled in none of the three tracks.
- $210$ candidates were enrolled in Artificial Intelligence.
- $180$ candidates were enrolled in Financial Technology.
- $150$ candidates were enrolled in Sustainable Energy.
- $90$ candidates were enrolled in both Artificial Intelligence and Financial Technology.
- $70$ candidates were enrolled in both Financial Technology and Sustainable Energy.
- $60$ candidates were enrolled in both Artificial Intelligence and Sustainable Energy.

How many candidates were enrolled in exactly one of the three tracks?

100

140

180

220

260

Cevabı ve açıklamayı göster

Cevap: $220$

Cevap

The number of candidates enrolled in exactly one of the three tracks is

220

.

To find the number of candidates enrolled in exactly one track, we first apply the Principle of Inclusion-Exclusion to find the triple intersection

|A \cap F \cap S|

:

|A \cup F \cup S| = |A| + |F| + |S| - (|A \cap F| + |F \cap S| + |A \cap S|) + |A \cap F \cap S|

. Substituting known values gives

360 = 210 + 180 + 150 - 220 + |A \cap F \cap S|

, so

|A \cap F \cap S| = 40

. Isolating mutually exclusive regions yields

100

candidates in Artificial Intelligence only,

60

in Financial Technology only, and

60

in Sustainable Energy only. Summing these gives

100 + 60 + 60 = 220

.

Adım Adım Çözüm

1

Determine the number of candidates enrolled in at least one track.

|A \cup F \cup S| = 400 - 40 = 360

Candidates in none of the tracks must be subtracted from the total survey population.

2

Calculate the number of candidates enrolled in all three tracks (

x = |A \cap F \cap S|

) using the Principle of Inclusion-Exclusion.

360 = 210 + 180 + 150 - (90 + 70 + 60) + x \implies 360 = 320 + x \implies x = 40

The sum of individual sets overcounts pairwise intersections and undercounts the triple intersection.

3

Calculate the counts for regions representing enrollment in exactly two tracks.

A \cap F \text{ only} = 90 - 40 = 50

,

F \cap S \text{ only} = 70 - 40 = 30

,

A \cap S \text{ only} = 60 - 40 = 20

Subtracting the triple intersection from each pairwise intersection isolates candidates enrolled in precisely two tracks.

4

Calculate the counts for candidates enrolled in exactly one track and sum them.

A \text{ only} = 210 - (50 + 20 + 40) = 100

;

F \text{ only} = 180 - (50 + 30 + 40) = 60

;

S \text{ only} = 150 - (20 + 30 + 40) = 60

. Total

= 100 + 60 + 60 = 220

.

Subtracting all overlap regions from each set total yields the single-track enrollment.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets

Soru 25Soru →

A cybersecurity firm audited a sample of $300$ corporate networks for compliance across three security standards: Network Encryption ( $E$ ), Multi-Factor Authentication ( $M$ ), and Access Logging ( $L$ ). The audit revealed that $160$ networks met standard $E$ , $140$ met standard $M$ , and $120$ met standard $L$ . Exactly $25$ networks met all three standards, while $45$ networks met none of the three standards. How many of the audited networks met exactly two of the three security standards?

90

115

140

165

190

Cevabı ve açıklamayı göster

Cevap: 115

Cevap

115 networks met exactly two of the three security standards.

The total number of networks meeting at least one standard is

300 - 45 = 255

. By the three-set inclusion-exclusion principle,

255 = 160 + 140 + 120 - S_2 + 25

, where

S_2

is the sum of the pairwise intersections

|E \cap M| + |M \cap L| + |E \cap L|

. Solving for

S_2

yields

S_2 = 190

. Since each pairwise intersection includes the 25 networks that met all three standards, the number of networks meeting exactly two standards is

190 - 3(25) = 115

.

Adım Adım Çözüm

1

Calculate the total number of networks that met at least one security standard.

|E \cup M \cup L| = 300 - 45 = 255

Subtracting networks that met none of the standards from the total sample size gives the union of the three sets.

2

Apply the Principle of Inclusion-Exclusion for three sets to find the sum of pairwise overlapping regions.

255 = 160 + 140 + 120 - (|E \cap M| + |M \cap L| + |E \cap L|) + 25

, so

|E \cap M| + |M \cap L| + |E \cap L| = 445 - 255 = 190

The formula

|E \cup M \cup L| = |E| + |M| + |L| - (|E \cap M| + |M \cap L| + |E \cap L|) + |E \cap M \cap L|

relates all known quantities.

3

Calculate the number of networks that met exactly two standards.

\text{Exactly 2} = (|E \cap M| + |M \cap L| + |E \cap L|) - 3|E \cap M \cap L| = 190 - 3(25) = 190 - 75 = 115

Each of the three pairwise intersections includes the triple intersection. Subtracting three times the triple intersection isolates the regions representing membership in exactly two sets.

Anahtar Kavram

Three-Set Principle of Inclusion-Exclusion

Soru 26Soru →

A community health center conducted a study of $250$ adults regarding their participation in three wellness programs: Nutrition Counseling ( $N$ ), Exercise Coaching ( $E$ ), and Stress Management ( $S$ ). The survey revealed the following data:

• $125$ adults participate in Nutrition Counseling.
• $105$ adults participate in Exercise Coaching.
• $85$ adults participate in Stress Management.
• $15$ adults participate in all three programs.
• $30$ adults participate in none of the three programs.

How many adults participate in exactly two of the three programs?

Cevabı ve açıklamayı göster

Cevap: 65

Cevap

65

The total number of surveyed adults is

250

, and

30

participate in none of the programs, meaning

220

adults participate in at least one program. Let

x_1

be the number of adults in exactly one program,

x_2

in exactly two programs, and

x_3 = 15

in all three programs. We have

x_1 + x_2 + 15 = 220

, which simplifies to

x_1 + x_2 = 205

. Additionally, summing the individual program participants yields

|N| + |E| + |S| = 125 + 105 + 85 = 315

. By region expansion,

|N| + |E| + |S| = x_1 + 2x_2 + 3x_3

. Substituting

x_3 = 15

gives

x_1 + 2x_2 + 45 = 315

, or

x_1 + 2x_2 = 270

. Subtracting

x_1 + x_2 = 205

from

x_1 + 2x_2 = 270

gives

x_2 = 65

.

Adım Adım Çözüm

1

Determine the number of adults in the union of all three set categories

|N \cup E \cup S| = 250 - 30 = 220

Subtracting the individuals participating in none of the programs from the total surveyed yields the total count of individuals participating in at least one program.

2

Set up an equation for the total unique participants using disjoint region variables

x_1 + x_2 = 205

The union equals

x_1 + x_2 + x_3 = 220

, where

x_1

represents adults in exactly 1 program,

x_2

in exactly 2, and

x_3 = 15

in all 3 programs.

3

Set up an equation using the sum of the individual program totals

x_1 + 2x_2 = 270

The sum

|N| + |E| + |S| = 125 + 105 + 85 = 315

counts single-program participants once, double-program participants twice, and triple-program participants three times (

x_1 + 2x_2 + 3(15) = 315

).

4

Solve the system of linear equations for

x_2

x_2 = 270 - 205 = 65

Subtracting

(x_1 + x_2 = 205)

from

(x_1 + 2x_2 = 270)

isolates

x_2

, which is the exact number of adults participating in exactly two programs.

Anahtar Kavram

3-Set Inclusion-Exclusion Principle & Venn Diagram Region Partitioning

Tahmini Süre:1m 45s

Soru 27Soru →

An aerospace engineering team evaluated telemetry logs from a constellation of $150$ satellites to monitor three types of sensor anomalies: Power Fluctuation ( $P$ ), Thermal Spike ( $T$ ), and Signal Attenuation ( $S$ ). The evaluation revealed the following:

- $65$ satellites exhibited Power Fluctuation.
- $58$ satellites exhibited Thermal Spike.
- $42$ satellites exhibited Signal Attenuation.
- $22$ satellites exhibited both Power Fluctuation and Thermal Spike.
- $18$ satellites exhibited both Thermal Spike and Signal Attenuation.
- $15$ satellites exhibited both Power Fluctuation and Signal Attenuation.
- $8$ satellites exhibited all three anomalies.

How many satellites exhibited exactly one of these three anomalies?

55

79

110

118

125

Cevabı ve açıklamayı göster

Cevap: 79

Cevap

79 satellites exhibited exactly one anomaly.

To find the number of satellites exhibiting exactly one anomaly, break down each set into its mutually exclusive regions. First, find the satellites in exactly two sets by subtracting the triple intersection (8) from each pairwise intersection: Power and Thermal only = 22 - 8 = 14; Thermal and Signal only = 18 - 8 = 10; Power and Signal only = 15 - 8 = 7. Next, subtract all overlapping regions from each individual set total: Power only = 65 - (14 + 7 + 8) = 36; Thermal only = 58 - (14 + 10 + 8) = 26; Signal only = 42 - (7 + 10 + 8) = 17. Adding these single-anomaly counts together gives 36 + 26 + 17 = 79.

Adım Adım Çözüm

1

Determine the number of satellites exhibiting ONLY pairwise anomalies (exactly two anomalies).

P and T only = 22 - 8 = 14; T and S only = 18 - 8 = 10; P and S only = 15 - 8 = 7.

The given pairwise intersection values include the 8 satellites that exhibited all three anomalies.

2

Calculate the number of satellites exhibiting exactly one anomaly for each set.

P only = 65 - (14 + 7 + 8) = 36; T only = 58 - (14 + 10 + 8) = 26; S only = 42 - (7 + 10 + 8) = 17.

Subtracting all elements that belong to two or three sets from each total set size leaves only those in the single set.

3

Sum the single-anomaly totals.

36 + 26 + 17 = 79.

The regions corresponding to exactly one anomaly are mutually exclusive.

Anahtar Kavram

Principle of Inclusion-Exclusion and Venn Diagram Region Decomposition

Soru 28Soru →

A software security audit evaluated $500$ open-source repositories for three specific vulnerability types: SQL Injection ( $S$ ), Cross-Site Scripting ( $X$ ), and Buffer Overflow ( $B$ ). Exactly $80$ repositories had none of these vulnerabilities. The audit revealed the following data:

- $220$ repositories contained SQL Injection vulnerabilities.
- $190$ repositories contained Cross-Site Scripting vulnerabilities.
- $210$ repositories contained Buffer Overflow vulnerabilities.
- $75$ repositories contained both SQL Injection and Cross-Site Scripting vulnerabilities.
- $80$ repositories contained both Cross-Site Scripting and Buffer Overflow vulnerabilities.
- $30$ repositories contained all three vulnerability types.

How many of the audited repositories contained Buffer Overflow vulnerabilities ONLY?

55

85

100

130

155

Cevabı ve açıklamayı göster

Cevap: $85$

Cevap

85 repositories contained Buffer Overflow vulnerabilities only.

The total number of repositories containing at least one vulnerability is

500 - 80 = 420

. Using the inclusion-exclusion formula

|S \cup X \cup B| = |S| + |X| + |B| - (|S \cap X| + |X \cap B| + |S \cap B|) + |S \cap X \cap B|

, we substitute the known values:

420 = 220 + 190 + 210 - (75 + 80 + |S \cap B|) + 30

. Solving gives

420 = 650 - 155 - |S \cap B|

, so

|S \cap B| = 75

. The repositories containing Buffer Overflow ONLY are given by subtracting the overlapping regions from the total Buffer Overflow set:

|B| - (|S \cap B| \text{ only}) - (|X \cap B| \text{ only}) - |S \cap X \cap B| = 210 - (75 - 30) - (80 - 30) - 30 = 210 - 45 - 50 - 30 = 85

.

Adım Adım Çözüm

1

Calculate the total number of repositories containing at least one vulnerability

|S \cup X \cup B| = 500 - 80 = 420

The total population is 500 and 80 repositories have no vulnerabilities.

2

Apply the 3-Set Inclusion-Exclusion Principle to determine the sum of pairwise intersections

|S \cap X| + |X \cap B| + |S \cap B| = 230

From

|S \cup X \cup B| = |S| + |X| + |B| - (|S \cap X| + |X \cap B| + |S \cap B|) + |S \cap X \cap B|

, we get

420 = 220 + 190 + 210 - \sum |\text{intersections}| + 30

, so

\sum |\text{intersections}| = 650 - 420 = 230

.

3

Find the missing pairwise intersection

|S \cap B|

|S \cap B| = 75

Since

|S \cap X| = 75

and

|X \cap B| = 80

, we have

75 + 80 + |S \cap B| = 230

, which yields

|S \cap B| = 75

.

4

Calculate the number of repositories with Buffer Overflow ONLY

|B \text{ only}| = 85

Subtract the exclusive two-set overlaps and the three-set overlap from

|B|

:

|B \text{ only}| = |B| - (|S \cap B| - |S \cap X \cap B|) - (|X \cap B| - |S \cap X \cap B|) - |S \cap X \cap B| = 210 - (75 - 30) - (80 - 30) - 30 = 210 - 45 - 50 - 30 = 85

.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Analysis

Soru 29Soru →

A survey of $280$ culinary school graduates evaluated their expertise across three specialized culinary disciplines: Molecular Gastronomy ( $M$ ), Pastry Arts ( $P$ ), and Sous-Vide Cooking ( $S$ ). The survey gathered the following data:
- $130$ graduates have expertise in Molecular Gastronomy.
- $140$ graduates have expertise in Pastry Arts.
- $120$ graduates have expertise in Sous-Vide Cooking.
- $50$ graduates have expertise in both Molecular Gastronomy and Pastry Arts.
- $45$ graduates have expertise in both Pastry Arts and Sous-Vide Cooking.
- $55$ graduates have expertise in both Molecular Gastronomy and Sous-Vide Cooking.
- $20$ graduates have expertise in all three disciplines.
- $20$ graduates have expertise in none of the three disciplines.

Which of the following statements MUST be true? Select all that apply.

Geçerli olan tümünü seçin

Exactly

45

graduates have expertise in Molecular Gastronomy only.

The total number of graduates with expertise in exactly two disciplines is

150

.

The number of graduates who have expertise in Sous-Vide Cooking but not Molecular Gastronomy is

65

.

More than

25\%

of all surveyed graduates have expertise in Pastry Arts only.

The total number of graduates who have expertise in at least one of the three disciplines is

240

.

Cevabı ve açıklamayı göster

Cevap: Exactly $45$ graduates have expertise in Molecular Gastronomy only.; The number of graduates who have expertise in Sous-Vide Cooking but not Molecular Gastronomy is $65$ .

Cevap

The statements confirming that exactly 45 graduates have expertise in Molecular Gastronomy only and that 65 graduates have expertise in Sous-Vide Cooking but not Molecular Gastronomy are correct.

The calculated region for Molecular Gastronomy only yields

130 - (30 + 35 + 20) = 45

, making the statement about Molecular Gastronomy only correct. Furthermore, the number of graduates with expertise in Sous-Vide Cooking but not Molecular Gastronomy equals the sum of the Sous-Vide only region (

40

) and the Pastry-Sous-Vide only region (

25

), which totals

65

, making that statement correct as well.

Adım Adım Çözüm

1

Determine the number of graduates in all three disciplines and in exactly two disciplines.

Triple intersection

|M \cap P \cap S| = 20

. Pairwise intersections excluding the triple intersection:

M \cap P \text{ only} = 50 - 20 = 30

,

P \cap S \text{ only} = 45 - 20 = 25

,

M \cap S \text{ only} = 55 - 20 = 35

. Total in exactly two disciplines =

30 + 25 + 35 = 90

.

Each given pairwise intersection contains the central triple intersection, so we must subtract 20 to find the exclusive double-overlap regions.

2

Calculate single-discipline exclusive counts.

Molecular Gastronomy only =

130 - (30 + 35 + 20) = 45

. Pastry Arts only =

140 - (30 + 25 + 20) = 65

. Sous-Vide Cooking only =

120 - (35 + 25 + 20) = 40

.

Subtracting all double-overlap and triple-overlap members from each set total yields the number of people belonging exclusively to that single set.

3

Evaluate the individual option statements.

Molecular Gastronomy only = 45 (True). Exactly two disciplines = 90 (Statement B claims 150, False). Sous-Vide Cooking without Molecular Gastronomy =

40 + 25 = 65

(True). Pastry Arts only percentage =

\frac{65}{280} \approx 23.21\% < 25\%

(Statement D claims

>25\%

, False). Total with at least one discipline =

280 - 20 = 260

(Statement E claims 240, False).

Comparing calculated region sizes against each option statement verifies which conditions must hold true.

Anahtar Kavram

3-Set Inclusion-Exclusion Principle and Venn Diagram Region Analysis

Tahmini Süre:2m 0s

Soru 30Soru →

A media analytics platform surveyed a group of $240$ subscribers regarding the genres of content they watched over the past month: Documentary ( $D$ ), Sci-Fi ( $S$ ), and Animation ( $A$ ). Among the surveyed subscribers:
- $110$ watched Documentary
- $120$ watched Sci-Fi
- $95$ watched Animation
- $45$ watched both Documentary and Sci-Fi
- $40$ watched both Sci-Fi and Animation
- $35$ watched both Documentary and Animation
- $15$ watched none of these three genres

How many of the surveyed subscribers watched content from exactly one of the three genres?

85

105

145

120

205

Cevabı ve açıklamayı göster

Cevap: 145

Cevap

145 subscribers watched content from exactly one genre.

The correct answer is 145. First, subtract the 15 subscribers who watched no genres from the total population of 240 to get 225 subscribers in the union of all three sets. Applying the principle of inclusion-exclusion yields

225 = 110 + 120 + 95 - (45 + 40 + 35) + |D \cap S \cap A|

, which solves to

|D \cap S \cap A| = 20

. With 20 subscribers watching all three genres, subscribers watching strictly two genres are 25 (Documentary & Sci-Fi only), 20 (Sci-Fi & Animation only), and 15 (Documentary & Animation only). Isolating subscribers watching only one genre gives

110 - (25 + 15 + 20) = 50

for Documentary,

120 - (25 + 20 + 20) = 55

for Sci-Fi, and

95 - (15 + 20 + 20) = 40

for Animation. Summing these single-genre counts (

50 + 55 + 40

) gives 145.

Adım Adım Çözüm

1

Find the number of subscribers who watched at least one genre and apply the 3-set Inclusion-Exclusion Principle to determine the number of subscribers who watched all three genres.

Number watching at least one genre:

240 - 15 = 225

. By inclusion-exclusion:

|D \cup S \cup A| = |D| + |S| + |A| - (|D \cap S| + |S \cap A| + |D \cap A|) + |D \cap S \cap A|

225 = 110 + 120 + 95 - (45 + 40 + 35) + |D \cap S \cap A|

225 = 325 - 120 + |D \cap S \cap A| = 205 + |D \cap S \cap A|

|D \cap S \cap A| = 225 - 205 = 20

Knowing the size of the central intersection region (

|D \cap S \cap A| = 20

) is necessary to isolate the individual non-overlapping Venn diagram regions.

2

Calculate the number of subscribers who watched EXCLUSIVELY two genres.

Documentary and Sci-Fi only:

45 - 20 = 25

Sci-Fi and Animation only:

40 - 20 = 20

Documentary and Animation only:

35 - 20 = 15

Subtracting the triple intersection from pairwise intersections isolates subscribers who belong to exactly two sets.

3

Calculate the number of subscribers who watched EXACTLY ONE genre and sum them.

Documentary only:

110 - (25 + 15 + 20) = 50

Sci-Fi only:

120 - (25 + 20 + 20) = 55

Animation only:

95 - (15 + 20 + 20) = 40

Total watching exactly one genre:

50 + 55 + 40 = 145

Subtracting all multi-set overlaps from each genre's total gives the unique single-genre viewers.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle and Regional Segmentation

Tahmini Süre:1m 45s

Set Theory Concepts and Venn Diagrams

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