Opening the book…
A gamers club at Baily Middle School consisting of 25 members was surveyed to find out who played board games, card games, or video games. Use the results depicted in the Venn diagram below to answer the following exercises.
How many gamers club members play all three types of games: board games, card games, and video games?
How many gamers are in the set Board \(\cap\) Video?
If Javier is in the region with a total of three members, what type of games does he play?
How many gamers play video games?
How many gamers are in the set Board \(\cup\) Card?
How many members of the gamers club do not play video games?
How many members of this club only play board games?
How many members of this club only play video games?
How many members of the gamers club play video and card games?
How many members of the gamers club are in the set Card'?
A blood drive at City Honors High School recently collected blood from 140 students, staff, and faculty. Use the results depicted in the Venn diagram below to answer the following exercises.
Blood type \(\mathrm{AB}^{+}\)is the universal acceptor. Of the 140 people who donated at City Honors, how many had blood type \(\mathrm{AB}^{+}\)?
Blood type \(\mathrm{O}^{-}\)is the universal donor. Anyone needing a blood transfusion can receive this blood type. How many people who donated blood during this drive had \(\mathrm{O}^{-}\)blood?
How many people donated with a type \(A\) blood factor?
How many people donated with a type \(A\) and type \(B\) blood factor (that is, they had type \(A B\) blood).
How many donors were \(\mathrm{O}^{+}\)?
How many donors were not \(\mathrm{Rh}^{+}\)?
Opal has blood type \(\mathrm{A}^{+}\). If she needs to have surgery that requires a blood transfusion, she can accept blood from anyone who does not have a type B blood factor. How many people donated blood during this drive at City Honors that Opal can accept?
Find \(n\left(A \cap \mathrm{Rh}^{+}\right)\).
Find \(n\left(A \cup \mathrm{Rh}^{+}\right)\).
Find \(n\left(A \cap B \cap \mathrm{Rh}^{-}\right)\).
For the following exercises, create a three circle Venn diagram to represent the relationship between the described sets.
The number of elements in the universal set, \(U\), is \(n(U)=48\). Sets \(A, B\), and \(C\) are subsets of \(U\) : \(n(A)=23, n(B)=25\), and \(n(C)=17\). Also, \(n(A \cap B)=15, n(B \cap C)=12, n(C \cap A)=11\), and \(n(A \cap B \cap C)=8\).
The number of elements in the universal set, \(U\), is \(n(U)=88\). Sets \(A, B\), and \(C\) are subsets of \(U\) : \(n(A)=31, n(B)=46: n(C)=33\). Also, \(n(A \cap B)=24, n(B \cap C)=24, n(C \cap A)=26\), and \(n(A \cap B \cap C)=22\).
The number of elements in the universal set, \(U\), is \(n(U)=52\). Sets \(A, B\), and \(C\) are subsets of \(U\) : \(n(A)=23, n(B)=27\), and \(n(C)=29\). Also, \(n(A \cap B)=22, n(B \cap C)=21, n(C \cap A)=19\), and \(n(A \cap B \cap C)=18\).
The number of elements in the universal set, \(U\), is \(n(U)=144\). Sets \(A, B\), and \(C\) are subsets of \(U\) : \(n(A)=36, n(B)=64\), and \(n(C)=81\). Also, \(n(A \cap B)=26, n(B \cap C)=61, n(C \cap A)=29\), and \(n(A \cap B \cap C)=25\).
The universal set, \(U\), has a cardinality of 36 . \(n(A)=12, n(B)=12, n(C)=15, n(A\) and \(B)=3, n(B\) and \(C)=4, n(C\) and \(A)=5\), \(n(C\) and \(A)=5\), and \(n(A\) and \(B\) and \(C)=1\).
The universal set, \(U\), has a cardinality of \(63 . n(A)=29\), \(n(B)=31, n(C)=41, n(A\) and \(B)=12, n(B\) and \(C)=16, n(C\) and \(A)=18\), and \(n(A\) and \(B\) and \(C)=5\).
The universal set, \(U\), has a cardinality of 72 . \(n(A)=32, n(B)=32, n(C)=44, n(A\) and \(B)=18, n(B\) and \(C)=22, n(C\) and \(A)=26\), and \(n(A\) and \(B\) and \(C)=14\).
The universal set, \(U\), has a cardinality of 81 . \(n(A)=54, n(B)=41, n(C)=52, n(A\) and \(B)=32, n(B\) and \(C)=28, n(C\) and \(A)=30\), and \(n(A\) and \(B\) and \(C)=21\).
The anime drawing club at Pratt Institute conducted a survey of its 42 members and found that 23 of them sketched with pastels, 28 used charcoal, and 17 used colored pencils. Of these, 10 club members used all three mediums, 18 used charcoal and pastels, 11 used colored pencils and charcoal, and 12 used colored pencils and pastels. The remaining club members did not use any of these three mediums.
A new SUV is selling with three optional packages: a sport package, a tow package, and an entertainment package. A dealership gathered the following data for all 31 of these vehicles sold during the month of July. A total of 18 SUVs included the entertainment package, 11 included the tow package, and 16 included the sport package. Of these, five SUVs included all three packages, seven were sold with both the tow package and sport package, 11 were sold with the entertainment and sport package, and eight were sold with the tow package and entertainment package. The remaining SUVs sold did not include any of these optional packages.
For the following exercises, perform the set operations as indicated on the following sets: \(U=\{\) red, orange, yellow, green, blue, indigo, violet \(\}, A=\{\) red, yellow, blue \(\}, B=\{\) orange, green, violet \(\}\), and \(C=\{\) red, green, indigo \(\}\).
Find \((A \cup B) \cap C\).
Find \((A \cap C) \cup B\).
Find \(U \cap(B \cup C)\).
Find \((B \cap A) \cap U\).
Find \(A \cap(B \cap C)^{\prime}\).
Find \(A^{\prime} \cap(B \cup C)\).
For the following exercises, perform the set operations as indicated on the following sets: \(U=\{20,21,22, \ldots, 29\}, A=\{21,24,27\}, B=\{20,22,24,28\}\), and \(C=\{21,23,25,27\}\).
Find \(A\) and \(B\) and \(C^{\prime}\).
Find \(A^{\prime}\) or \(B\) or \(C\).
Find \((A\) or \(B)\) and \(C^{\prime}\).
Find \((A\) or \(B)\) or \(C^{\prime}\).
Find \((A\) and \(C)\) and \(B^{\prime}\).
Find \((A \text { or } B)^{\prime}\) and \(C\).
For the following exercises, use Venn diagrams to prove the following properties of sets:
Commutative property for the union of two sets: \(A \cup B=B \cup A\).
Commutative property for the intersection of two sets: \(A \cap B=B \cap A\).
Associative property for the intersection of three sets: \((A \cap B) \cap C=A \cap(B \cap C)\).
Associative property for the union of three sets: \(A \cup(B \cup C)=(A \cup B) \cup C\).
Distributive property for set intersection over set union: \(A \cap(B \cup C)=(A \cap B) \cup(A \cap C)\).
Distributive property for set union over set intersection: \(A \cup(B \cap C)=(A \cup B) \cap(A \cup C)\).
1.5.0: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.
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