Visualizing Data with Venn Diagrams: Problem-Solving in Set Theory

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You need to write out every step of your calculation including Venn Diagram, not just the final answer, in order to receive full credit.

(If you don’t show work, you will get 55% by default).

Q#1: Construct Venn diagram : U=1000

n(A)=520, n(B)=330 & n(A⋂B) =199

Help: Venn Diagram with two sets

Q#2: Construct Venn diagram: U=2550

n(A)=225, n(B)=410, n(C)=356

n(A⋂B) =56, n(B⋂C) =90, n(A⋂C) =110 and n(A⋂B⋂C) = 17

Help:  How to Construct Venn Diagram with Three Sets?

Q#3: A college survey was taken to determine which social media platform students use. Of 538 students​ surveyed, 95 used

Facebook​, 78 used Snap-chat​, 35 used both the Facebook and Snap-chat. Draw a Venn Diagram then answer the

following;

a. How many used Facebook only?

b. How many used Snap-chat only?

c. How many Facebook or Snap-chat?

d. How many Snap-chat but not Facebook?

e. How many did not used either the Facebook or the Snap-chat?

Help: Solving Application Problems of 2 Sets With Venn Diagram

Q#4: In a survey of​ four-year colleges, it was found that 318 offered a criminal justice degree. 319 offered Information technology degree. 557 offered a nursing degree. 95 offered a criminal justice degree and Information technology degree.103 offered criminal justice degree and a nursing degree. 95 offered Information technology degree and a nursing degree. 29 offered a criminal justice, Information technology degree, and a nursing degree. 35 offered none of these degrees.

a. How many is criminal justice degree only?

b. How many​ four-year colleges were​ surveyed?

c. A criminal justice degree and a nursing​ degree, but not IT degree?

d. IT degree, but neither a criminal justice degree nor a nursing​ degree?

e. A criminal justice, IT​ degree, and a nursing ​degree?

Help: Solving Venn diagram application problem by three sets

Q#5: Using the data below construct a Venn diagram then answer the following questions.

survey of faculty and graduate students at the University of Hollywood ‘s film school

revealed the following information:

51 admire Moe

49 admire Larry

60 admire Curly

34 admire Moe and Larry

32 admire Larry and Curly

36 admire Moe and Curly

24 admire all three of the Stooges

1 admires none of the Three Stooges

a. How many people were surveyed?

b. How many admire Moe, but not Larry and Curly?

Help: Solving Venn diagram application problem by three sets

Struggling with where to start this assignment? Follow this guide to tackle your assignment easily!

Q#1: Construct a Venn Diagram for Two Sets

Given Data:

• Universal set: $U=1000$
• $n(A)=520$ (total in set A)
• $n(B)=330$ (total in set B)
• $n(A\cap B)=199$ (in both sets A and B)

Step 1: Find the numbers in only A, only B, and neither

1. Only in A: $$n(A\text{ only})=n(A)-n(A\cap B)=520-199=321$$
2. Only in B: $$n(B\text{ only})=n(B)-n(A\cap B)=330-199=131$$
3. Neither A nor B: $$n(\text{Neither})=U-(n(A\text{ only})+n(B\text{ only})+n(A\cap B))$$ $$=1000-(321+131+199)=1000-651=349$$

Final Values for Venn Diagram

• Only A = 321
• Only B = 131
• Both A and B = 199
• Neither A nor B = 349

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Q#2: Construct a Venn Diagram for Three Sets

Given Data:

• $U=2550$
• $n(A)=225,n(B)=410,n(C)=356$
• $n(A\cap B)=56,n(B\cap C)=90,n(A\cap C)=110$
• $n(A\cap B\cap C)=17$

Step 1: Find the values for individual sections

1. Only A: $$n(A\text{ only})=n(A)-(n(A\cap B)+n(A\cap C)-n(A\cap B\cap C))$$ $$=225-(56+110-17)=225-149=76$$
2. Only B: $$n(B\text{ only})=n(B)-(n(A\cap B)+n(B\cap C)-n(A\cap B\cap C))$$ $$=410-(56+90-17)=410-129=281$$
3. Only C: $$n(C\text{ only})=n(C)-(n(A\cap C)+n(B\cap C)-n(A\cap B\cap C))$$ $$=356-(110+90-17)=356-183=173$$
4. Neither A, B, nor C: $$n(\text{Neither})=U-(n(A\text{ only})+n(B\text{ only})+n(C\text{ only})+n(A\cap B)+n(B\cap C)+n(A\cap C)-2n(A\cap B\cap C))$$ $$=2550-(76+281+173+56+90+110-2(17))$$ $$=2550-766=1784$$

Final Values for Venn Diagram

• Only A = 76
• Only B = 281
• Only C = 173
• A & B only = 56 – 17 = 39
• B & C only = 90 – 17 = 73
• A & C only = 110 – 17 = 93
• A & B & C = 17
• Neither = 1784

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Q#3: College Survey on Social Media Use

Given Data:

• $U=538$
• $n(Facebook)=95$
• $n(Snapchat)=78$
• $n(Facebook\cap Snapchat)=35$

Step 1: Find the values for individual sections

1. Facebook Only: $$n(FB\text{ only})=95-35=60$$
2. Snapchat Only: $$n(Snap\text{ only})=78-35=43$$
3. Neither: $$n(\text{Neither})=U-(n(FB\text{ only})+n(Snap\text{ only})+n(Facebook\cap Snapchat))$$ $$=538-(60+43+35)=538-138=400$$

Final Answers

• (a) Facebook Only = 60
• (b) Snapchat Only = 43
• (c) Facebook or Snapchat = 138
• (d) Snapchat but not Facebook = 43
• (e) Neither = 400

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Q#4: College Degrees

Given Data:

• $n(CJ)=318,n(IT)=319,n(N)=557$
• $n(CJ\cap IT)=95,n(CJ\cap N)=103,n(IT\cap N)=95$
• $n(CJ\cap IT\cap N)=29$
• $n(\text{None})=35$

Final Answers

1. CJ Only = $318-(95+103-29)=149$
2. Total Colleges Surveyed = $318+319+557-95-103-95+29+35=865$
3. CJ & Nursing but not IT = $103-29=74$
4. IT Only = $319-(95+95-29)=158$
5. All Three = $29$

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Q#5: Admiration of Three Stooges

Given Data:

• $n(M)=51,n(L)=49,n(C)=60$
• $n(M\cap L)=34,n(L\cap C)=32,n(M\cap C)=36$
• $n(M\cap L\cap C)=24$
• $n(\text{None})=1$

Step 1: Find the values for individual sections

1. Only Moe: $$n(M\text{ only})=51-(34+36-24)=51-46=5$$

Final Answers

• (a) Total People Surveyed = $51+49+60-34-32-36+24+1=123$
• (b) Moe Only = 5

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