How to interpret Venn diagrams - SSAT Upper Level Quantitative
Card 1 of 40
The above represents a Venn diagram. The universal set 
is the set of all positive integers.
Let 
represent the set of multiples of 7; let 
represent all of the multiples of 11; let 
represent all of the multiples of 13. As you can see, the three sets divide the universal set into eight regions. Suppose each positive integer was placed in the correct region. Which of the following numbers would be in the same region as 2,431?
The above represents a Venn diagram. The universal set
Let
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The region in which 2,431 appears depends on the sets of which 2,431 is an element, which in turn depends on which of 7, 11, and 13 divides it evenly:
2,431 is a multiple of 11 and 13, but not 7, so 2,431 is in 
and 
, but not 
. We look for a number among the choices that is in 
and 
, but not 
- that is, a number divisible by 11 and 13 but not 7.
, so 2,772 is divisible by 7. We can eliminate it.
, so 2,184 is divisible by 7. We can eliminate it.
, so 3,081 is not divisible by 11, and we can eliminate it.
, so 2,409 is not divisible by 13, and we can eliminate it.
However:
2,145 is divisible by 11 and 13 but not 7, so this is the correct choice.
The region in which 2,431 appears depends on the sets of which 2,431 is an element, which in turn depends on which of 7, 11, and 13 divides it evenly:
2,431 is a multiple of 11 and 13, but not 7, so 2,431 is in
However:
2,145 is divisible by 11 and 13 but not 7, so this is the correct choice.
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Given the Venn diagram below, which of the following does not belong to 
?
Given the Venn diagram below, which of the following does not belong to
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The symbol 
stands for the union between two sets. Therefore, 
means the set of all numbers that are in either A or B. Looking at our choices, the only number that isn't in either A, B, or both is 23.
The symbol
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The universal set 
represented by the above Venn diagram is the set of all natural numbers from 
to 
inclusive.
The subsets are:
: The set of all multiples of 
: The set of all multiples of 
: The set of all multiples of 
How many elements are in the set represented by the shaded region?
The universal set
The subsets are:
How many elements are in the set represented by the shaded region?
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The shaded region is 
, which will comprise all of the numbers that are multiples of 10 and of either or both 8 or 9.
will comprise multiples of 10 and 8 - that is, multiples of 
. Since 
,
will comprise multiples of 10 and 9 - that is, multiples of 
. Since 
,
To find 
, we first find 
.
, the set of numbers that are multiples of 8, 9, and 10. 
, so we look for multiples of 360, of which there are two under 1,000 (360 and 720).
The shaded region is
To find
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The universal set 
represented by the above Venn diagram is the set of all natural numbers from 1 to 1,000 inclusive.
The subsets are:
: The set of all multiples of 
: The set of all multiples of 
: The set of all multiples of 
How many elements are in the set represented by the shaded region?
The universal set
The subsets are:
How many elements are in the set represented by the shaded region?
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The shaded region is inside 
and 
, and outside of 
, meaning that the shaded set represents
Examine 
first. Each number in 
must be a multiple of 7 and 8; since the two are relatively prime, each number is a multiple of 56.
so seventeen elements are in 
.
We eliminate all elements in 
- that is, all elements that also multiples of 6. These elements are 168, 336, 504, 672, 840 - a total of five.
This leaves twelve elements in 
.
The shaded region is inside
Examine
so seventeen elements are in
We eliminate all elements in
This leaves twelve elements in
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The above Venn diagram represents all of this year's graduating seniors at Rockwell High School, the universal set 
.
represents all of the students who are in the National Honor Society.
represents all of the students who became old enough to vote in the November 5 election during their senior year.
represents all of the students who enrolled in a French course during senior year.
Cathy was inducted into the National Honor Society in her junior year, and is still a member. She turned 18 on January 4 during her senior year, and she is carrying a respectable B average in her school's third-year French course. If her name were to be written in the above diagram in the correct place, in which of the five numbered regions would her name fall?
The above Venn diagram represents all of this year's graduating seniors at Rockwell High School, the universal set
Cathy was inducted into the National Honor Society in her junior year, and is still a member. She turned 18 on January 4 during her senior year, and she is carrying a respectable B average in her school's third-year French course. If her name were to be written in the above diagram in the correct place, in which of the five numbered regions would her name fall?
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Cathy is in the Honor Society, meaning that she is in set 
; she turned 18 after election day, so she is not in set 
; she is taking a French course, so she is in set 
. She is in set 
, represented by region 2.
Cathy is in the Honor Society, meaning that she is in set
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In the above Venn diagram, the universal set 
is the set of Presidents of the United States.
represents the set of Presidents who were born after 1850.
represents the set of Presidents who were born in a state completely west of the Mississippi River.
represents the set of Presidents who served eight years or more.
Which of the following Presidents would fall in the pink region?
In the above Venn diagram, the universal set
Which of the following Presidents would fall in the pink region?
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The shaded region is inside set 
, so we are looking for a President who was born after 1850; this eliminates Lincoln.
The region is outside of 
, so we want a President who served fewer than eight years. This eliminates Wilson and Reagan.
The region is outside of 
, so we want the state of the President's birth to be fully or partly east of the Mississippi River. This eliminates Nixon.
The correct response is Harding.
The shaded region is inside set
The region is outside of
The region is outside of
The correct response is Harding.
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A group of high school juniors are taking Biology, Calculus, and Spanish as shown above. Which student is not in the set 
?
A group of high school juniors are taking Biology, Calculus, and Spanish as shown above. Which student is not in the set
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The notation 
stands for "union," which refers to everything that is in either set. 
refers to the group of students taking either Calculus or Spanish (everyone on this diagram except those taking only Biology). From the diagram, Patrick and Ashley are the only students taking neither Calculus nor Spanish, so Patrick is the correct answer.
The notation
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Giving the Venn diagram above, what is the sum of the numbers in the set 
?
Giving the Venn diagram above, what is the sum of the numbers in the set
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The notation 
stands for "A union C," which refers to everything that is in either set 
or set 
.
When we add the numbers together, we get:
The notation
When we add the numbers together, we get:
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Examine the above Venn diagram. The universal set 
is defined to be 
, with each element placed in its correct region in the diagram. What is 
?
Examine the above Venn diagram. The universal set
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is the complement of 
, the set of all elements in the universal set not in 
. In the diagram, it is represented by all elements not inside the circle that represents 
. This is the set:
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Examine the above Venn diagram. The universal set 
is defined to be 
, with each element placed in its correct region in the diagram. What is 
?
Examine the above Venn diagram. The universal set
Tap to reveal answer
is the union of the sets 
and 
- that is, the set of all elements in either 
or 
. 
is the complement of set 
- that is, the set of elements in 
not in 
; 
is defined similarly. Therefore, we need all of the elements either outside of 
or outside of 
. The excluded elements will be those inside both sets, which, by the diagram, can be seen to be 3, 5, 13, and 14. Therefore,
.
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The above represents a Venn diagram. The universal set is the set of all positive integers.
Let represent the set of multiples of 7; let represent all of the multiples of 11; let represent all of the multiples of 13. As you can see, the three sets divide the universal set into eight regions. Suppose each positive integer was placed in the correct region. Which of the following numbers would be in the same region as 2,431?
The above represents a Venn diagram. The universal set
Let
Tap to reveal answer
The region in which 2,431 appears depends on the sets of which 2,431 is an element, which in turn depends on which of 7, 11, and 13 divides it evenly:
2,431 is a multiple of 11 and 13, but not 7, so 2,431 is in and , but not . We look for a number among the choices that is in and , but not - that is, a number divisible by 11 and 13 but not 7.
, so 2,772 is divisible by 7. We can eliminate it.
, so 2,184 is divisible by 7. We can eliminate it.
, so 3,081 is not divisible by 11, and we can eliminate it.
, so 2,409 is not divisible by 13, and we can eliminate it.
However:
2,145 is divisible by 11 and 13 but not 7, so this is the correct choice.
The region in which 2,431 appears depends on the sets of which 2,431 is an element, which in turn depends on which of 7, 11, and 13 divides it evenly:
2,431 is a multiple of 11 and 13, but not 7, so 2,431 is in
However:
2,145 is divisible by 11 and 13 but not 7, so this is the correct choice.
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Given the Venn diagram below, which of the following does not belong to ?
Given the Venn diagram below, which of the following does not belong to
Tap to reveal answer
The symbol stands for the union between two sets. Therefore, means the set of all numbers that are in either A or B. Looking at our choices, the only number that isn't in either A, B, or both is 23.
The symbol
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The universal set represented by the above Venn diagram is the set of all natural numbers from to inclusive.
The subsets are:
: The set of all multiples of
: The set of all multiples of
: The set of all multiples of
How many elements are in the set represented by the shaded region?
The universal set
The subsets are:
How many elements are in the set represented by the shaded region?
Tap to reveal answer
The shaded region is , which will comprise all of the numbers that are multiples of 10 and of either or both 8 or 9.
will comprise multiples of 10 and 8 - that is, multiples of . Since ,
will comprise multiples of 10 and 9 - that is, multiples of . Since ,
To find , we first find .
, the set of numbers that are multiples of 8, 9, and 10. , so we look for multiples of 360, of which there are two under 1,000 (360 and 720).
The shaded region is
To find
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The universal set represented by the above Venn diagram is the set of all natural numbers from 1 to 1,000 inclusive.
The subsets are:
: The set of all multiples of
: The set of all multiples of
: The set of all multiples of
How many elements are in the set represented by the shaded region?
The universal set
The subsets are:
How many elements are in the set represented by the shaded region?
Tap to reveal answer
The shaded region is inside and , and outside of , meaning that the shaded set represents
Examine first. Each number in must be a multiple of 7 and 8; since the two are relatively prime, each number is a multiple of 56.
so seventeen elements are in .
We eliminate all elements in - that is, all elements that also multiples of 6. These elements are 168, 336, 504, 672, 840 - a total of five.
This leaves twelve elements in .
The shaded region is inside
Examine
so seventeen elements are in
We eliminate all elements in
This leaves twelve elements in
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The above Venn diagram represents all of this year's graduating seniors at Rockwell High School, the universal set .
represents all of the students who are in the National Honor Society.
represents all of the students who became old enough to vote in the November 5 election during their senior year.
represents all of the students who enrolled in a French course during senior year.
Cathy was inducted into the National Honor Society in her junior year, and is still a member. She turned 18 on January 4 during her senior year, and she is carrying a respectable B average in her school's third-year French course. If her name were to be written in the above diagram in the correct place, in which of the five numbered regions would her name fall?
The above Venn diagram represents all of this year's graduating seniors at Rockwell High School, the universal set
Cathy was inducted into the National Honor Society in her junior year, and is still a member. She turned 18 on January 4 during her senior year, and she is carrying a respectable B average in her school's third-year French course. If her name were to be written in the above diagram in the correct place, in which of the five numbered regions would her name fall?
Tap to reveal answer
Cathy is in the Honor Society, meaning that she is in set ; she turned 18 after election day, so she is not in set ; she is taking a French course, so she is in set . She is in set , represented by region 2.
Cathy is in the Honor Society, meaning that she is in set
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In the above Venn diagram, the universal set is the set of Presidents of the United States.
represents the set of Presidents who were born after 1850.
represents the set of Presidents who were born in a state completely west of the Mississippi River.
represents the set of Presidents who served eight years or more.
Which of the following Presidents would fall in the pink region?
In the above Venn diagram, the universal set
Which of the following Presidents would fall in the pink region?
Tap to reveal answer
The shaded region is inside set , so we are looking for a President who was born after 1850; this eliminates Lincoln.
The region is outside of , so we want a President who served fewer than eight years. This eliminates Wilson and Reagan.
The region is outside of , so we want the state of the President's birth to be fully or partly east of the Mississippi River. This eliminates Nixon.
The correct response is Harding.
The shaded region is inside set
The region is outside of
The region is outside of
The correct response is Harding.
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Giving the Venn diagram above, what is the sum of the numbers in the set ?
Giving the Venn diagram above, what is the sum of the numbers in the set
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The notation stands for "A union C," which refers to everything that is in either set or set .
When we add the numbers together, we get:
The notation
When we add the numbers together, we get:
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A group of high school juniors are taking Biology, Calculus, and Spanish as shown above. Which student is not in the set ?
A group of high school juniors are taking Biology, Calculus, and Spanish as shown above. Which student is not in the set
Tap to reveal answer
The notation stands for "union," which refers to everything that is in either set. refers to the group of students taking either Calculus or Spanish (everyone on this diagram except those taking only Biology). From the diagram, Patrick and Ashley are the only students taking neither Calculus nor Spanish, so Patrick is the correct answer.
The notation
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Examine the above Venn diagram. The universal set is defined to be , with each element placed in its correct region in the diagram. What is ?
Examine the above Venn diagram. The universal set
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is the complement of , the set of all elements in the universal set not in . In the diagram, it is represented by all elements not inside the circle that represents . This is the set:
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Examine the above Venn diagram. The universal set is defined to be , with each element placed in its correct region in the diagram. What is ?
Examine the above Venn diagram. The universal set
Tap to reveal answer
is the union of the sets and - that is, the set of all elements in either or . is the complement of set - that is, the set of elements in not in ; is defined similarly. Therefore, we need all of the elements either outside of or outside of . The excluded elements will be those inside both sets, which, by the diagram, can be seen to be 3, 5, 13, and 14. Therefore,
.
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