How to use a Venn Diagram - ISEE Upper Level: Quantitative Reasoning
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In a school of 
students, 
students take Greek, 
take Old English, and 
take neither. How many take both?
In a school of
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Based on the information given, you can construct the following Venn Diagram:
In order to find the overlap, you need to find out how many are in the circles together. This is easy. Subtract: 
. Now, since the overlap represents a duplication, you need to subtract out one of those duplicate values. Let's call that 
; therefore, we know that:
Solving for 
, you get:
Based on the information given, you can construct the following Venn Diagram:
In order to find the overlap, you need to find out how many are in the circles together. This is easy. Subtract:
Solving for
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In a group of 
people, 
have a laptop and 
have a tablet. Of those people who have a laptop or a tablet, 
have both. How many people in the total group have neither a laptop nor a tablet?
In a group of
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Based on the information given, you can draw the following Venn Diagram:
To solve this, remember that the total number of values in the two circles is:
(We must do this because of the overlap. You need to subtract out one instance of that overlap.)
If we assign the value 
for the unknown region, we know:
Based on the information given, you can draw the following Venn Diagram:
To solve this, remember that the total number of values in the two circles is:
(We must do this because of the overlap. You need to subtract out one instance of that overlap.)
If we assign the value
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In a group of plants, 
are green, 
have large leaves, and 
are both green and have large leaves. How many plants are green without large leaves?
In a group of plants,
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Based on the information, you can draw the following Venn Diagram:
It is very easy to solve for the number of plants that have green leaves but not large ones. This is merely 
. We find this by eliminating the large-leaved plants from the green ones (by subtracting the overlap from the green ones).
Based on the information, you can draw the following Venn Diagram:
It is very easy to solve for the number of plants that have green leaves but not large ones. This is merely
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In a group of 
people, 
have books, 
have pens, and 
have neither books nor pens. How many people in the group have only books?
In a group of
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Based on the information given, you can draw the following Venn Diagram:
Now, you must begin by solving for 
. You know that the two circles together will have 
in them. This is arrived at by subtracting the people who have neither books nor pens (
) from the "universe" of people in the sample space (
). Now, we know that 
. This is because of the overlap of 
in both groups. We have to get rid of one instance of that. Thus we can solve for 
:
Now, we can find the number of people with only books by subtracting 
from the 
to get 
.
Based on the information given, you can draw the following Venn Diagram:
Now, you must begin by solving for
Now, we can find the number of people with only books by subtracting
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Examine the above Venn diagram. Let 
be the universal set of the Presidents of the United States. 
is the set of all Presidents born in Virginia; 
is the set of all Presidents born after 1850; 
is the set of all Presidents whose first name was or is James.
James Abram Garfield was born in Ohio in 1831. In which region would he fall?
Examine the above Venn diagram. Let
James Abram Garfield was born in Ohio in 1831. In which region would he fall?
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Carter would not fall in set A, since he was not a President born in Virginia.
He would not fall in B, since he was born before 1850.
He would fall in C, since his first name is James.
He would fall in the region included in set C, but not A or B - this is Region V.
Carter would not fall in set A, since he was not a President born in Virginia.
He would not fall in B, since he was born before 1850.
He would fall in C, since his first name is James.
He would fall in the region included in set C, but not A or B - this is Region V.
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Examine the above Venn diagram. Let 
be the universal set of the Presidents of the United States. 
is the set of all Presidents born in Virginia; 
is the set of all Presidents born after 1850; 
is the set of all Presidents whose first name was or is James.
James Earl Carter was born in Georgia in 1924. In which region would he fall?
Examine the above Venn diagram. Let
James Earl Carter was born in Georgia in 1924. In which region would he fall?
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Carter would not fall in set A, since he was not a President born in Virginia.
He would fall in B, since he was born after 1850.
He would fall in C, since his first name is James.
He would fall in the region included in sets B and C, but not A - this is Region III.
Carter would not fall in set A, since he was not a President born in Virginia.
He would fall in B, since he was born after 1850.
He would fall in C, since his first name is James.
He would fall in the region included in sets B and C, but not A - this is Region III.
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Examine the above Venn diagram. Let universal set 
represent the set of all words in the English language.
Let 
be the set of all words whose last letter is a vowel. Let 
be the set of all words whose first letter is a consonant. Let 
be the set of all words exactly six letters in length.
Which of the following would be a subset of the set represented by the shaded region in the diagram?
Note: for purposes of this question, "Y" is considered a consonant.
Examine the above Venn diagram. Let universal set
Let
Which of the following would be a subset of the set represented by the shaded region in the diagram?
Note: for purposes of this question, "Y" is considered a consonant.
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The subset must comprise words that fall inside sets 
and 
, but not 
. Therefore, all of the words in the subset must begin with a consonant, end with a vowel, and not have six letters.
Of the given choices, the only set whose elements fit this description is {plateau, portmanteau, calliope, marionette, taco}.
The subset must comprise words that fall inside sets
Of the given choices, the only set whose elements fit this description is {plateau, portmanteau, calliope, marionette, taco}.
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Examine the above Venn diagram. Let universal set 
represent the set of all words in the English language.
Let 
be the set of all words whose last letter is a consonant. Let 
be the set of all words whose first letter is a vowel. Let 
be the set of all words exactly five letters in length.
Which of the following would be a subset of the set represented by the shaded region in the diagram?
Note: for purposes of this question, "Y" is considered a consonant.
Examine the above Venn diagram. Let universal set
Let
Which of the following would be a subset of the set represented by the shaded region in the diagram?
Note: for purposes of this question, "Y" is considered a consonant.
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The subset must comprise words that fall inside set 
, but neither
nor 
.
Therefore, all of the words in the subset must have exactly five letters, but cannot begin with a vowel or end with a consonant - that is, we are looking for a set of five-letter words that begin with a consonant and end with a vowel.
The only set among the five choices that matches this description is the set
{price, value, pinna, trove, three}.
The subset must comprise words that fall inside set
Therefore, all of the words in the subset must have exactly five letters, but cannot begin with a vowel or end with a consonant - that is, we are looking for a set of five-letter words that begin with a consonant and end with a vowel.
The only set among the five choices that matches this description is the set
{price, value, pinna, trove, three}.
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The following Venn diagram depicts the number of students who play hockey, football, and baseball. How many students play football and baseball?
The following Venn diagram depicts the number of students who play hockey, football, and baseball. How many students play football and baseball?
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The number of students who play football or baseball can by finding the summer of the number of students who play football alone, baseball alone, baseball and football, and all three sports.
The number of students who play football or baseball can by finding the summer of the number of students who play football alone, baseball alone, baseball and football, and all three sports.
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A class of 
students was asked whether they have cats, dogs, or both.The results are depicted in the following Venn diagram. How many students do not have a dog?
A class of
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First, calculate the number of students with a dog:
Next, subtract the number of students with a dog from the total number of students.
First, calculate the number of students with a dog:
Next, subtract the number of students with a dog from the total number of students.
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Let the universal set 
be the set of all positive integers. Define:
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
Let the universal set
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
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The grayed portion of the Venn diagram corresponds to those integers which are not in any of 
, 
, or 
. Therefore, we eliminate any choices that are in any of the three sets.
is the set of integers which end in 1 or 6; we can eliminate 166 and 176 immediately.
is the set of perfect square integers; we can eliminate 144, since 
.
is the set of integers which, when divided by 4, yields remainder 2. Since 
, we can eliminate 154.
All four choices have been eliminated.
The grayed portion of the Venn diagram corresponds to those integers which are not in any of
All four choices have been eliminated.
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In the above Venn diagram, the universal set is defined as 
. Each of the eight letters is placed in its correct region. Which of the following is equal to 
?
In the above Venn diagram, the universal set is defined as
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is the complement of 
- the set of all elements in 
not in 
.
is the union of sets 
and 
- the set of all elements in either 
or 
. Therefore, 
is the set of all elements in neither 
nor 
, which can be seen from the diagram to be only one element - 
. Therefore,
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Let the universal set be the set of all positive integers. Define:
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
Let the universal set
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
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The grayed portion of the Venn diagram corresponds to those integers which are not in any of , , or . Therefore, we eliminate any choices that are in any of the three sets.
is the set of integers which end in 1 or 6; we can eliminate 166 and 176 immediately.
is the set of perfect square integers; we can eliminate 144, since .
is the set of integers which, when divided by 4, yields remainder 2. Since , we can eliminate 154.
All four choices have been eliminated.
The grayed portion of the Venn diagram corresponds to those integers which are not in any of
All four choices have been eliminated.
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Refer to the above Venn diagram.
Define universal set 
, the set of natural numbers.
Define sets 
and 
as follows:
Which of the following numbers is an element of the set represented by the gray area in the diagram?
Refer to the above Venn diagram.
Define universal set
Define sets
Which of the following numbers is an element of the set represented by the gray area in the diagram?
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The gray area represents the set of all elements that are in 
but not in 
.
is the set of integers that, when divided by 3, yield remainder 2. Therefore, we can eliminate 102 and 105, both multiples of 3, and 103, which, when divided by 3, yields remainder 1.
is the set of integers that, when divided by 4, yield remainder 1. Since we do not want an element from this set, we can eliminate 101, but not 104.
104 is the correct choice.
The gray area represents the set of all elements that are in
104 is the correct choice.
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Let the universal set be the set of all positive integers. Define:
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
Let the universal set
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
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The grayed portion of the Venn diagram corresponds to those integers which are not in any of , , or . Therefore, we eliminate any choices that are in any of the three sets.
is the set of integers which end in 1 or 6; we can eliminate 166 and 176 immediately.
is the set of perfect square integers; we can eliminate 144, since .
is the set of integers which, when divided by 4, yields remainder 2. Since , we can eliminate 154.
All four choices have been eliminated.
The grayed portion of the Venn diagram corresponds to those integers which are not in any of
All four choices have been eliminated.
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Refer to the above Venn diagram.
Define universal set , the set of natural numbers.
Define sets and as follows:
Which of the following numbers is an element of the set represented by the gray area in the diagram?
Refer to the above Venn diagram.
Define universal set
Define sets
Which of the following numbers is an element of the set represented by the gray area in the diagram?
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The gray area represents the set of all elements that are in but not in .
is the set of integers that, when divided by 3, yield remainder 2. Therefore, we can eliminate 102 and 105, both multiples of 3, and 103, which, when divided by 3, yields remainder 1.
is the set of integers that, when divided by 4, yield remainder 1. Since we do not want an element from this set, we can eliminate 101, but not 104.
104 is the correct choice.
The gray area represents the set of all elements that are in
104 is the correct choice.
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Refer to the above Venn diagram.
Define universal set , the set of natural numbers.
Define sets and as follows:
Which of the following numbers is an element of the set represented by the gray area in the diagram?
Refer to the above Venn diagram.
Define universal set
Define sets
Which of the following numbers is an element of the set represented by the gray area in the diagram?
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The gray area represents the set of all elements that are in but not in .
is the set of integers that, when divided by 3, yield remainder 2. Therefore, we can eliminate 102 and 105, both multiples of 3, and 103, which, when divided by 3, yields remainder 1.
is the set of integers that, when divided by 4, yield remainder 1. Since we do not want an element from this set, we can eliminate 101, but not 104.
104 is the correct choice.
The gray area represents the set of all elements that are in
104 is the correct choice.
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Let the universal set be the set of all positive integers. Define:
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
Let the universal set
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
Tap to reveal answer
The grayed portion of the Venn diagram corresponds to those integers which are not in any of , , or . Therefore, we eliminate any choices that are in any of the three sets.
is the set of integers which end in 1 or 6; we can eliminate 166 and 176 immediately.
is the set of perfect square integers; we can eliminate 144, since .
is the set of integers which, when divided by 4, yields remainder 2. Since , we can eliminate 154.
All four choices have been eliminated.
The grayed portion of the Venn diagram corresponds to those integers which are not in any of
All four choices have been eliminated.
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Refer to the above Venn diagram.
Define universal set , the set of natural numbers.
Define sets and as follows:
Which of the following numbers is an element of the set represented by the gray area in the diagram?
Refer to the above Venn diagram.
Define universal set
Define sets
Which of the following numbers is an element of the set represented by the gray area in the diagram?
Tap to reveal answer
The gray area represents the set of all elements that are in but not in .
is the set of integers that, when divided by 3, yield remainder 2. Therefore, we can eliminate 102 and 105, both multiples of 3, and 103, which, when divided by 3, yields remainder 1.
is the set of integers that, when divided by 4, yield remainder 1. Since we do not want an element from this set, we can eliminate 101, but not 104.
104 is the correct choice.
The gray area represents the set of all elements that are in
104 is the correct choice.
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In the above Venn diagram, the universal set is defined as . Each of the eight letters is placed in its correct region. Which of the following is equal to ?
In the above Venn diagram, the universal set is defined as
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is the complement of - the set of all elements in not in .
is the union of sets and - the set of all elements in either or . Therefore, is the set of all elements in neither nor , which can be seen from the diagram to be only one element - . Therefore,
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