Overlapping Sets and Inclusion-Exclusion Principle Alıştırma Soruları — GMAT

Soru 1Soru →

A clinical study evaluated $150$ patient records for the presence of three specific health biomarkers: Biomarker X, Biomarker Y, and Biomarker Z. The study revealed that $70$ patients had Biomarker X, $60$ patients had Biomarker Y, and $50$ patients had Biomarker Z. Exactly $35$ patients had exactly two of these biomarkers, and exactly $10$ patients had all three biomarkers. How many of the patients evaluated had none of the three biomarkers?

5

15

25

35

45

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Cevap: 25

Cevap

25 patients evaluated had none of the three biomarkers.

By the inclusion-exclusion principle for three sets, the total number of elements in at least one set is given by

|X \cup Y \cup Z| = |X| + |Y| + |Z| - (\text{Elements in exactly two sets}) - 2(\text{Elements in all three sets})

. Substituting the given values yields

70 + 60 + 50 - 35 - 2(10) = 180 - 35 - 20 = 125

. Since there are

150

total patients evaluated, the number of patients with none of the three biomarkers is

150 - 125 = 25

.

Adım Adım Çözüm

1

Sum the individual counts for all three biomarker groups.

\text{Sum of individual sets} = 70 + 60 + 50 = 180

This sum counts individuals with 1 biomarker once, individuals with 2 biomarkers twice, and individuals with 3 biomarkers three times.

2

Apply the inclusion-exclusion formula adapted for 'exactly two' and 'all three' overlapping subsets.

\text{At least one biomarker} = 180 - (\text{Exactly two}) - 2 \times (\text{All three}) = 180 - 35 - 2(10) = 125

To count each person with at least one biomarker exactly once, we subtract the count of patients with exactly two biomarkers once, and the count of patients with all three biomarkers twice.

3

Subtract the number of patients with at least one biomarker from the total population of patients evaluated.

\text{None} = 150 - 125 = 25

The total group consists of patients with at least one biomarker plus patients with none of the three biomarkers.

Anahtar Kavram

Three-Set Overlapping Sets and Inclusion-Exclusion Principle

Tahmini Süre:1m 45s

Soru 2Soru →

In a technology firm of $100$ software engineers, $65$ engineers know Python and $45$ know Java. If $20$ engineers know both Python and Java, how many of the engineers know neither Python nor Java?

10

0

15

20

30

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

The number of engineers who know neither Python nor Java is

10

.

The total number of engineers is

100

. By inclusion-exclusion, the union of Python and Java speakers is

65 + 45 - 20 = 90

. The number of engineers who know neither language is the total minus the union,

100 - 90 = 10

.

Adım Adım Çözüm

1

Calculate the number of engineers who know at least one language using the Inclusion-Exclusion Principle.

\text{At least one} = \text{Python} + \text{Java} - \text{Both} = 65 + 45 - 20 = 90

Engineers who know both languages are counted in both individual totals, so their count must be subtracted once to avoid double counting.

2

Subtract the number of engineers who know at least one language from the total number of engineers to find those who know neither.

\text{Neither} = \text{Total} - \text{At least one} = 100 - 90 = 10

The total group consists of those who know at least one language plus those who know neither.

Anahtar Kavram

Two-Set Inclusion-Exclusion Principle

Tahmini Süre:45s

Soru 3Soru →

At a financial analytics firm, a group of 200 analysts were evaluated on their proficiency in three software tools: Options Analytics, Futures Trader, and Swaps Pricing. Exactly 15% of the analysts had no proficiency in any of the three tools. Among the remaining analysts, 110 were proficient in Options Analytics, 95 were proficient in Futures Trader, and 85 were proficient in Swaps Pricing. If exactly 20 analysts were proficient in all three tools, how many analysts were proficient in exactly one of the three tools?

10

30

50

70

100

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

70 analysts were proficient in exactly one of the three tools.

The total number of analysts is 200, and 15% (30 analysts) have no proficiency, leaving 170 analysts proficient in at least one tool. Applying the 3-set inclusion-exclusion formula

N(A \cup B \cup C) = N(A) + N(B) + N(C) - S_2 + N(A \cap B \cap C)

, we get

170 = 110 + 95 + 85 - S_2 + 20

, which yields

S_2 = 140

. Since

S_2

counts elements in exactly two sets once and elements in all three sets three times, the number of analysts proficient in exactly two tools is

140 - 3(20) = 80

. Finally, subtracting those proficient in exactly two tools (80) and all three tools (20) from the total proficient in at least one tool (170) gives

170 - 80 - 20 = 70

.

Adım Adım Çözüm

1

Calculate the total number of analysts proficient in at least one tool

\text{At least one} = 200 - (0.15 \times 200) = 200 - 30 = 170

Analysts who are not proficient in any tool must be excluded from the total group size to find the union of the three sets.

2

Apply the 3-set inclusion-exclusion formula to find the sum of pairwise intersections

170 = 110 + 95 + 85 - S_2 + 20 \implies 170 = 310 - S_2 \implies S_2 = 140

, where

S_2 = N(\text{Options} \cap \text{Futures}) + N(\text{Futures} \cap \text{Swaps}) + N(\text{Options} \cap \text{Swaps})

The standard inclusion-exclusion principle states that

N(A \cup B \cup C) = N(A) + N(B) + N(C) - S_2 + N(A \cap B \cap C)

.

3

Determine the number of analysts proficient in exactly two tools

\text{Exactly 2} = S_2 - 3 \times N(\text{All 3}) = 140 - 3(20) = 140 - 60 = 80

Each member of the triple intersection is counted 3 times in

S_2

. Subtracting

3 \times N(\text{All 3})

isolates the elements belonging to exactly two sets.

4

Calculate the number of analysts proficient in exactly one tool

\text{Exactly 1} = N(\text{At least 1}) - \text{Exactly 2} - N(\text{All 3}) = 170 - 80 - 20 = 70

The union of the three sets consists of elements proficient in exactly 1 tool, exactly 2 tools, and all 3 tools.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle and Subset Decomposition

Tahmini Süre:2m 0s

Soru 4Soru →

At an international conference of $200$ delegates, every delegate speaks at least one of three languages: French, German, or Spanish. Exactly $115$ delegates speak French, $85$ speak German, and $67$ speak Spanish. Furthermore, exactly $15$ delegates speak all three languages. If the number of delegates who speak only one language is four times the number of delegates who speak exactly two languages, how many delegates speak exactly two of these three languages?

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Cevap: 37

Cevap

The number of delegates who speak exactly two of the three languages is 37.

By dividing the 200 delegates into three disjoint groups—those speaking exactly 1 language (

E_1

), exactly 2 languages (

E_2

), and all 3 languages (

E_3 = 15

)—we establish that

E_1 + E_2 + 15 = 200

, giving

E_1 + E_2 = 185

. Given

E_1 = 4E_2

, substituting yields

5E_2 = 185

, which gives

E_2 = 37

.

Adım Adım Çözüm

1

Define exact-count variables for the overlapping sets

Let

E_1

be the number of delegates speaking exactly one language,

E_2

be the number speaking exactly two languages, and

E_3 = 15

be the number speaking all three languages.

Categorizing elements into mutually exclusive subsets (exactly 1, exactly 2, exactly 3) simplifies 3-set inclusion-exclusion problems.

2

Set up the total population equation

E_1 + E_2 + E_3 = 200 \implies E_1 + E_2 + 15 = 200 \implies E_1 + E_2 = 185

Since every delegate speaks at least one language, the sum of all mutually exclusive regions equals the total number of delegates.

3

Apply the given relationship between

E_1

and

E_2

Substitute

E_1 = 4E_2

into

E_1 + E_2 = 185

to get

4E_2 + E_2 = 185 \implies 5E_2 = 185

The problem establishes that the number of single-language speakers is four times the number of dual-language speakers.

4

Solve for the target quantity

E_2

E_2 = 37

Dividing 185 by 5 yields the exact count of delegates speaking exactly two languages.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle using Exact-Overlap Subsets

Soru 5Soru →

A market research firm surveyed $300$ consumers regarding their subscriptions to three streaming services: Service X, Service Y, and Service Z.

- $40$ consumers subscribed to none of the three services.
- $160$ consumers subscribed to Service X.
- $140$ consumers subscribed to Service Y.
- $130$ consumers subscribed to Service Z.
- Exactly $60$ consumers subscribed to exactly two of the services.

How many consumers subscribed to all three services?

35

50

55

65

110

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Cevap: 55

Cevap

55 consumers subscribed to all three services.

The total number of consumers subscribing to at least one service is

300 - 40 = 260

. Expressing the total union in terms of non-overlapping region counts gives

e_1 + e_2 + e_3 = 260

, where

e_1

is the number subscribing to exactly one service,

e_2 = 60

is the number subscribing to exactly two, and

e_3

is the number subscribing to all three. Thus,

e_1 + e_3 = 200

. Summing the individual totals gives

|X| + |Y| + |Z| = e_1 + 2e_2 + 3e_3 = 430

, which simplifies to

e_1 + 3e_3 = 310

. Subtracting

e_1 + e_3 = 200

from

e_1 + 3e_3 = 310

yields

2e_3 = 110

, so

e_3 = 55

.

Adım Adım Çözüm

1

Calculate the total number of consumers who subscribed to at least one streaming service.

Total in union

|X \cup Y \cup Z| = 300 - 40 = 260

.

Consumers subscribing to at least one service represent the total sample minus those subscribing to none.

2

Set up the region equations for 3-set inclusion-exclusion in terms of elements in exactly 1 set (

e_1

), exactly 2 sets (

e_2

), and all 3 sets (

e_3

).

Equation 1:

e_1 + e_2 + e_3 = 260

; Equation 2:

|X| + |Y| + |Z| = e_1 + 2e_2 + 3e_3 = 160 + 140 + 130 = 430

.

Summing individual set sizes counts elements in exactly one set once, elements in exactly two sets twice, and elements in all three sets three times.

3

Substitute

e_2 = 60

into both equations.

Equation 1:

e_1 + 60 + e_3 = 260 \implies e_1 + e_3 = 200

; Equation 2:

e_1 + 2(60) + 3e_3 = 430 \implies e_1 + 3e_3 = 310

.

Simplifying both equations yields a system of two linear equations in terms of

e_1

and

e_3

.

4

Subtract the simplified Equation 1 from Equation 2 to solve for

e_3

.

(e_1 + 3e_3) - (e_1 + e_3) = 310 - 200 \implies 2e_3 = 110 \implies e_3 = 55

.

Subtracting eliminates

e_1

, directly yielding the number of consumers subscribing to all three services.

Anahtar Kavram

3-Set Inclusion-Exclusion Principle using Exact Region Counts

Soru 6Soru →

A research group evaluated 150 renewable energy projects. Each project utilizes at least one of three primary energy technologies: solar, wind, or hydro. Exactly 85 projects utilize solar power, 60 projects utilize wind power, and 45 projects utilize hydro power. Furthermore, 20 projects utilize both solar and wind power, 15 projects utilize both solar and hydro power, and 10 projects utilize both wind and hydro power. How many of the surveyed projects utilize all three energy technologies?

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Cevap: 5

Cevap

5 projects utilize all three energy technologies.

Applying the three-set inclusion-exclusion principle Total = |A| + |B| + |C| - (|A ∩ B| + |A ∩ C| + |B ∩ C|) + |A ∩ B ∩ C| + Neither gives 150 = 85 + 60 + 45 - (20 + 15 + 10) + |A ∩ B ∩ C| + 0. Simplifying yields 150 = 145 + |A ∩ B ∩ C|, which gives |A ∩ B ∩ C| = 5.

Adım Adım Çözüm

1

Identify known set sizes and recall the Principle of Inclusion-Exclusion for three sets.

Total = 150, |S| = 85, |W| = 60, |H| = 45, |S ∩ W| = 20, |S ∩ H| = 15, |W ∩ H| = 10, Neither = 0.

Setting up known quantities ensures proper application of set intersection rules.

2

Substitute the set sizes into the formula: Total = |S| + |W| + |H| - (|S ∩ W| + |S ∩ H| + |W ∩ H|) + |S ∩ W ∩ H| + Neither.

150 = 85 + 60 + 45 - (20 + 15 + 10) + |S ∩ W ∩ H| + 0.

Pairwise overlaps are double-counted when individual sets are summed and must be subtracted; the central triple overlap is over-subtracted and must be added back.

3

Combine known terms and isolate the unknown triple intersection.

150 = 190 - 45 + |S ∩ W ∩ H| => 150 = 145 + |S ∩ W ∩ H| => |S ∩ W ∩ H| = 5.

Subtracting 145 from 150 yields the exact count of projects belonging to all three sets.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle

Tahmini Süre:2m 0s

Soru 7Soru →

A museum cataloged $250$ historical artifacts. Each artifact was subjected to at least one of three preservation procedures: micro-scanning, chemical analysis, or radiocarbon dating. Exactly $140$ artifacts underwent micro-scanning, $120$ underwent chemical analysis, and $100$ underwent radiocarbon dating. If $40$ artifacts underwent both micro-scanning and chemical analysis, $35$ underwent both chemical analysis and radiocarbon dating, and $45$ underwent both micro-scanning and radiocarbon dating, how many artifacts underwent all three procedures?

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Cevap: 10

Cevap

10 artifacts underwent all three preservation procedures.

Using the Inclusion-Exclusion Principle for three sets, the total union is equal to the sum of individual set sizes (

140 + 120 + 100 = 360

), minus the sum of two-set overlaps (

40 + 35 + 45 = 120

), plus the three-set overlap. Setting up

250 = 360 - 120 + x

yields

250 = 240 + x

, so

x = 10

.

Adım Adım Çözüm

1

Set up the Inclusion-Exclusion formula for three overlapping sets.

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

The total size of the union equals the sum of the individual set sizes minus the pairwise intersections plus the triple intersection.

2

Substitute the given values into the formula.

250 = 140 + 120 + 100 - (40 + 35 + 45) + |A \cap B \cap C|

Since every artifact underwent at least one procedure, the total number of artifacts equals the union of all three sets.

3

Compute the sum of individual sets and pairwise intersections to isolate the unknown.

250 = 360 - 120 + |A \cap B \cap C| \Rightarrow 250 = 240 + |A \cap B \cap C|

Simplifying the arithmetic terms allows direct calculation of the triple intersection.

4

Subtract

240

from

250

to find the final value.

|A \cap B \cap C| = 10

Solving the linear equation yields the number of artifacts undergoing all three procedures.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets

Soru 8Soru →

A corporate venture capital firm evaluated $120$ technology startups for investment. Each startup met at least one of three key criteria: strong artificial intelligence capability, established revenue growth, or international market presence. Exactly $65$ startups met the artificial intelligence criteria, $55$ met the revenue growth criteria, and $50$ met the international presence criteria. Furthermore, $25$ startups met both the artificial intelligence and revenue growth criteria, $20$ met both the revenue growth and international presence criteria, and $15$ met both the artificial intelligence and international presence criteria. How many startups met all three investment criteria?

5

10

15

20

25

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

The number of startups that met all three investment criteria is 10.

According to the Inclusion-Exclusion Principle for three sets, the total population is equal to the sum of the three individual sets minus the sum of the three pairwise intersections, plus the intersection of all three sets. Substituting the given values:

120 = 65 + 55 + 50 - (25 + 20 + 15) + x

, which simplifies to

120 = 110 + x

, giving

x = 10

. Thus,

10

startups met all three criteria.

Adım Adım Çözüm

1

State 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|

To account for startups counted in multiple overlapping categories without double-counting or over-subtracting.

2

Substitute the known values from the problem statement into the formula

120 = 65 + 55 + 50 - (25 + 20 + 15) + x

, where

x = |A \cap B \cap C|

Every startup meets at least one criterion, so the union size equals the total number of startups (

120

).

3

Simplify the sums and solve for

x

120 = 170 - 60 + x \implies 120 = 110 + x \implies x = 10

Subtracting

110

from

120

gives the exact number of startups meeting all three criteria.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle

Overlapping Sets and Inclusion-Exclusion Principle

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