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Questions and Answers
Given three sets A, B, and C within a universal set U, which expression correctly represents the elements that are in A or B, but not in C?
Given three sets A, B, and C within a universal set U, which expression correctly represents the elements that are in A or B, but not in C?
- $(A \cup B) \cap C'$ (correct)
- $(A \cup B) \cap C$
- $(A \cap B) \cup C'$
- $(A \cup B) \cup C'$
Suppose a survey asks people if they like apples (A), bananas (B), or cherries (C). If $n(A) = 50$, $n(B) = 60$, $n(C) = 70$, $n(A \cap B) = 25$, $n(B \cap C) = 30$, $n(A \cap C) = 20$, and $n(A \cap B \cap C) = 10$, what is $n(A \cup B \cup C)$?
Suppose a survey asks people if they like apples (A), bananas (B), or cherries (C). If $n(A) = 50$, $n(B) = 60$, $n(C) = 70$, $n(A \cap B) = 25$, $n(B \cap C) = 30$, $n(A \cap C) = 20$, and $n(A \cap B \cap C) = 10$, what is $n(A \cup B \cup C)$?
- 100
- 105
- 115 (correct)
- 95
In a Venn diagram representing sets A, B, and C, the region corresponding to $A \cap (B \cup C)$ represents which of the following?
In a Venn diagram representing sets A, B, and C, the region corresponding to $A \cap (B \cup C)$ represents which of the following?
- Elements that are in A or in B or in C
- Elements that are in A and also in both B and C
- Elements that are in A, B, and C
- Elements that are in A and either in B or in C, or in both (correct)
If $n(U) = 200$, $n(A) = 100$, $n(B) = 80$, and $n(A \cup B) = 150$, what is $n(A' \cap B')$?
If $n(U) = 200$, $n(A) = 100$, $n(B) = 80$, and $n(A \cup B) = 150$, what is $n(A' \cap B')$?
Consider a Venn diagram with three sets: X, Y, and Z. Which of the following expressions represents the area containing elements that are exclusively in X and Y, but not in Z?
Consider a Venn diagram with three sets: X, Y, and Z. Which of the following expressions represents the area containing elements that are exclusively in X and Y, but not in Z?
Given three sets A, B, and C, what is the simplified form of $(A \cup B) \cap (A \cup C)$?
Given three sets A, B, and C, what is the simplified form of $(A \cup B) \cap (A \cup C)$?
In a survey of 100 people, 40 like apples, 30 like bananas, and 20 like both. How many people like neither apples nor bananas?
In a survey of 100 people, 40 like apples, 30 like bananas, and 20 like both. How many people like neither apples nor bananas?
If A and B are two sets and $A \subseteq B$, then what is $A \cup B$?
If A and B are two sets and $A \subseteq B$, then what is $A \cup B$?
Given sets A and B, if $P(A) = 0.6$, $P(B) = 0.5$, and $P(A \cap B) = 0.3$, what is $P(A \cup B)$?
Given sets A and B, if $P(A) = 0.6$, $P(B) = 0.5$, and $P(A \cap B) = 0.3$, what is $P(A \cup B)$?
In a Venn diagram, the area outside both circles A and B represents which set operation?
In a Venn diagram, the area outside both circles A and B represents which set operation?
If $U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$, $A = {1, 3, 5, 7, 9}$, and $B = {2, 4, 6, 8, 10}$, what is $A \cap B$?
If $U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$, $A = {1, 3, 5, 7, 9}$, and $B = {2, 4, 6, 8, 10}$, what is $A \cap B$?
Given $U = {1, 2, 3, ..., 10}$, $A = {1, 2, 3, 4, 5}$, and $B = {4, 5, 6, 7, 8}$. What is $(A - B) \cup (B - A)$?
Given $U = {1, 2, 3, ..., 10}$, $A = {1, 2, 3, 4, 5}$, and $B = {4, 5, 6, 7, 8}$. What is $(A - B) \cup (B - A)$?
In the context of conditional probability and Venn diagrams, what does $P(A|B)$ represent?
In the context of conditional probability and Venn diagrams, what does $P(A|B)$ represent?
If $P(A) = 0.4$, $P(B) = 0.6$, and A and B are independent events, what is $P(A \cap B)$?
If $P(A) = 0.4$, $P(B) = 0.6$, and A and B are independent events, what is $P(A \cap B)$?
Given three sets A, B, and C, how is the region representing elements that are in A and B, but not in C, expressed?
Given three sets A, B, and C, how is the region representing elements that are in A and B, but not in C, expressed?
Let $A = {x : x \in \mathbb{Z}, 0 < x < 5}$ and $B = {x : x \in \mathbb{Z}, 2 \le x < 7}$. What is $A \cup B$?
Let $A = {x : x \in \mathbb{Z}, 0 < x < 5}$ and $B = {x : x \in \mathbb{Z}, 2 \le x < 7}$. What is $A \cup B$?
What is the complement of the universal set U?
What is the complement of the universal set U?
If $n(A) = 20$, $n(B) = 30$, and $n(A \cup B) = 40$, what is $n(A \cap B)$?
If $n(A) = 20$, $n(B) = 30$, and $n(A \cup B) = 40$, what is $n(A \cap B)$?
Which of the following is equivalent to $(A \cap B)'$?
Which of the following is equivalent to $(A \cap B)'$?
Consider three sets A, B, and C. How would you represent the elements that are in A or B, but not in C?
Consider three sets A, B, and C. How would you represent the elements that are in A or B, but not in C?
If $P(A) = 0.7$ and $P(B|A) = 0.4$, what is $P(A \cap B)$?
If $P(A) = 0.7$ and $P(B|A) = 0.4$, what is $P(A \cap B)$?
What does the expression $(A - B) \cup (B - A)$ represent?
What does the expression $(A - B) \cup (B - A)$ represent?
Let A, B, and C be sets. What is the expression for elements that belong to A and B, but do not belong to C?
Let A, B, and C be sets. What is the expression for elements that belong to A and B, but do not belong to C?
In survey results, 70% like apples, 60% like bananas, and 40% like both. What percentage likes either apples or bananas?
In survey results, 70% like apples, 60% like bananas, and 40% like both. What percentage likes either apples or bananas?
Given sets $A = {1, 2, 3, 4, 5}$ and $B = {3, 4, 5, 6, 7}$, what is the symmetric difference between A and B, denoted as $A \triangle B$?
Given sets $A = {1, 2, 3, 4, 5}$ and $B = {3, 4, 5, 6, 7}$, what is the symmetric difference between A and B, denoted as $A \triangle B$?
If $n(U) = 50$, $n(A) = 30$, $n(B) = 25$, and $n(A \cap B) = 10$, what is $n((A \cup B)') $?
If $n(U) = 50$, $n(A) = 30$, $n(B) = 25$, and $n(A \cap B) = 10$, what is $n((A \cup B)') $?
Given two events A and B, where $P(A) = 0.5$, $P(B) = 0.6$, and $P(A \cup B) = 0.8$, what is $P(A \cap B)$?
Given two events A and B, where $P(A) = 0.5$, $P(B) = 0.6$, and $P(A \cup B) = 0.8$, what is $P(A \cap B)$?
In a class of 50 students, 20 play cricket and 30 play football. If 10 students play both, how many play neither?
In a class of 50 students, 20 play cricket and 30 play football. If 10 students play both, how many play neither?
What is the correct representation of the area that includes elements exclusive to set A or set B in a Venn diagram?
What is the correct representation of the area that includes elements exclusive to set A or set B in a Venn diagram?
Flashcards
Venn Diagrams
Venn Diagrams
Diagrams showing logical relations between a finite collection of different sets.
Set
Set
A collection of distinct objects, treated as a single entity.
Elements of a Set
Elements of a Set
Objects contained within a set.
Universal Set
Universal Set
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Subset
Subset
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Empty Set
Empty Set
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Union of Sets (A ∪ B)
Union of Sets (A ∪ B)
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Intersection of Sets (A ∩ B)
Intersection of Sets (A ∩ B)
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Complement of a Set (A')
Complement of a Set (A')
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Difference of Sets (A - B)
Difference of Sets (A - B)
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Venn Diagram Representation
Venn Diagram Representation
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Using Venn Diagrams
Using Venn Diagrams
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Two-Set Venn Diagrams
Two-Set Venn Diagrams
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Three-Set Venn Diagrams
Three-Set Venn Diagrams
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Conditional Probability in Venn Diagrams
Conditional Probability in Venn Diagrams
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Key Formula for Union of Two Sets
Key Formula for Union of Two Sets
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Key Formula for Union of Three Sets
Key Formula for Union of Three Sets
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Probability of Union of Two Events
Probability of Union of Two Events
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Probability of Complement
Probability of Complement
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Study Notes
- Venn diagrams illustrate logical relationships between a finite collection of sets
Basic Set Theory
- A set is a collection of distinct objects treated as a single entity
- Elements are the individual objects within a set
- x ∈ A denotes that 'x' is an element of set A
- x ∉ A denotes that 'x' is not an element of set A
- The universal set 'U' or 'ξ' encompasses all possible elements under consideration
- A subset (A ⊆ B) exists when all elements of A are also found in B
- A proper subset (A ⊂ B) occurs when A ⊆ B and A ≠B
- The empty set (∅ or {}) contains no elements and is a subset of every set
Set Operations
- The union of sets A and B (A ∪ B) includes all elements present in A, B, or both
- The intersection of sets A and B (A ∩ B) contains elements common to both A and B
- The complement of set A (A' or Ac) comprises all elements in the universal set U that are not in A
- The difference of sets A and B (A - B or A \ B) contains elements in A but not in B
Venn Diagram Representation
- Sets are represented by circles (or shapes) inside a rectangle (the universal set) in Venn diagrams
- Overlapping areas between circles signify the intersection of sets
- The area outside a circle represents the complement of that set
Using Venn Diagrams
- Venn diagrams provide a visual representation of sets and their relationships
- Enable understanding and solving problems in set theory, logic, and probability domains
- Each region in a Venn diagram represents a specific combination of set membership
- Determine the number of elements within sets and their combinations
Two-Set Venn Diagrams
- Two overlapping circles within a rectangle form a two-set Venn diagram
- The rectangle symbolizes the universal set, U
- Each circle represents a set, A and B
- The overlapping region is A ∩ B
- The region in A but not in B is A - B
- The region in B but not in A is B - A
- The region outside both circles is (A ∪ B)', equal to A' ∩ B'
Three-Set Venn Diagrams
- Three overlapping circles within a rectangle form a three-set Venn diagram
- Each circle represents a set: A, B, and C
- Overlapping regions represent intersections: A ∩ B, A ∩ C, B ∩ C, and A ∩ B ∩ C
- Each region signifies a unique combination of set membership
- Problems involving three sets are solved by populating each region with the number of elements
Problem Solving with Venn Diagrams
- Start by carefully reading the problem to define sets and the universal set
- Draw a Venn diagram with the correct number of circles
- Populate each region with element counts, beginning with the intersection of all sets (e.g., A ∩ B ∩ C) if available
- Deduce the number of elements in other regions using provided information
- Progress outwards from intersections to individual sets
- Utilize set operations (union, intersection, complement, difference) to determine quantities
Conditional Probability in Venn Diagrams
- Visualize conditional probability using Venn diagrams
- P(A|B) represents the probability of A occurring, given that B has occurred
- P(A|B) = P(A ∩ B) / P(B)
- P(A|B) in a Venn diagram is the proportion of set B that is also in set A
Key Formulae
- n(A ∪ B) = n(A) + n(B) - n(A ∩ B), where n(X) represents the number of elements in set X
- n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- P(A') = 1 - P(A)
Common Mistakes
- Forgetting to subtract the intersection when calculating set unions
- Misinterpreting Venn diagram regions
- Incorrect application of set operations
- Neglecting the universal set when calculating complements
- Double-counting elements in overlapping regions
Example Problem 1
- In a class of 30 students, 18 play football, 15 play basketball, and 8 play both
- Draw a Venn diagram to represent this scenario
- Find the amount of students who play neither football, nor basketball
- Let F = students who play football, and B = students who play basketball
- n(F) = 18, n(B) = 15, n(F ∩ B) = 8
- n(F ∪ B) = n(F) + n(B) - n(F ∩ B) = 18 + 15 - 8 = 25
- The amount of students who play neither is the total subtracted by n(F ∪ B) = 30 - 25 = 5
Example Problem 2
- In a survey, 60 people were asked if they read newspaper A, newspaper B, or newspaper C
- 25 read A, 26 read B, 26 read C, 9 read both A and B, 11 read both B and C, 8 read both A and C, and 3 read all three newspapers
- Find out how many people read none of the newspapers
- n(A) = 25, n(B) = 26, n(C) = 26, n(A ∩ B) = 9, n(B ∩ C) = 11, n(A ∩ C) = 8, n(A ∩ B ∩ C) = 3
- n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)
- n(A ∪ B ∪ C) = 25 + 26 + 26 - 9 - 11 - 8 + 3 = 52
- The amount of people who read none of the newspapers = Total people - n(A ∪ B ∪ C) = 60 - 52 = 8
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