DSQ: Understanding sets - GMAT Quantitative
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In the above Venn diagram, universal set 
represents the residents of Jonesville. The sets 
represent the set of all Toastmasters, Elks, and Masons, respectively.
Jerry is a resident of Jonesville. Is he a Mason?
Statement 1: Jerry is a Toastmaster.
Statement 2: Jerry is not an Elk.
In the above Venn diagram, universal set
Jerry is a resident of Jonesville. Is he a Mason?
Statement 1: Jerry is a Toastmaster.
Statement 2: Jerry is not an Elk.
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The question is equivalent to asking whether Jerry is an element of set 
.
The sets 
and 
are disjoint - they have no elements in common. From Statement 1 alone, Jerry is an element of 
, so he cannot be an element of 
. He is not a Mason.
From Statement 2 alone, Jerry is an element of 
. Since there are elements not in 
that are and are not elements of 
, it cannot be determined whether Jerry is an element of 
- a Mason.
The question is equivalent to asking whether Jerry is an element of set
The sets
From Statement 2 alone, Jerry is an element of
In the above Venn diagram, universal set 
represents the residents of a city. The sets 
represent the set of all teenagers, even birth month, and males, respectively.
Jamie is a resident of this city. Does Jamie have an even birth month?
Statement 1: Jamie is not a male.
Statement 2: Jamie is not a teenager.
In the above Venn diagram, universal set
Jamie is a resident of this city. Does Jamie have an even birth month?
Statement 1: Jamie is not a male.
Statement 2: Jamie is not a teenager.
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The question is whether or not Jamie is an element of 
.
Assume both statements to be true. Then Jamie is an element of the set 
, shaded in this Venn diagram:
Some elements of 
are elements of 
, but some are not, making the two statements together insufficient to answer the question of whether Jamie is an element of . Whether Jamie was born in an even month or not cannot be determined.
The question is whether or not Jamie is an element of
Assume both statements to be true. Then Jamie is an element of the set
Some elements of
In the above Venn diagram, universal set 
represents the residents of Eastland. The sets 
represent the set of all Toastmasters, Elks, and Masons, respectively.
Craig is a resident of Eastland. Is Craig a Toastmaster?
Statement 1: Craig is not a Mason.
Statement 2: Craig is not an Elk.
In the above Venn diagram, universal set
Craig is a resident of Eastland. Is Craig a Toastmaster?
Statement 1: Craig is not a Mason.
Statement 2: Craig is not an Elk.
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The question is whether or not Craig is an element of 
.
Assume both statements to be true. Craig is an element of the set 
, shaded in this Venn diagram:
There are elements of this set that both are and are not elements of 
. Therefore, the two statements together do not prove or disprove Craig to be an element of 
, a Toastmaster.
The question is whether or not Craig is an element of
Assume both statements to be true. Craig is an element of the set
There are elements of this set that both are and are not elements of
In the above Venn diagram, universal set 
represents the residents of Jacksonville. The sets 
represent the set of all Toastmasters, Elks, and Masons, respectively.
Jimmy is a resident of Jacksonville. Is Jimmy a Mason?
Statement 1: Jimmy is not a Toastmaster.
Statement 2: Jimmy is not an Elk.
In the above Venn diagram, universal set
Jimmy is a resident of Jacksonville. Is Jimmy a Mason?
Statement 1: Jimmy is not a Toastmaster.
Statement 2: Jimmy is not an Elk.
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The question asks whether Jimmy is an element of 
.
Statement 1 alone - that Jimmy is an element of 
- provides insufficient information, since 
contains elements that are and are not elements of 
. By a similar argument, Statement 2 alone is insufficient.
Now assume both statements to be true. Then Jimmy is an element of 
, shaded in the Venn diagram below:
It can be seen that shares no elements with 
, so Jimmy cannot be an element of 
. Jimmy is not a Mason.
The question asks whether Jimmy is an element of
Statement 1 alone - that Jimmy is an element of
Now assume both statements to be true. Then Jimmy is an element of
It can be seen that
Two of the courses from which the 98 freshmen at a high school may choose are French and Creative Writing.
How many freshmen enrolled in neither course?
Statement 1: 10 freshmen enrolled in both courses.
Statement 2: 21 freshmen enrolled in each course.
Two of the courses from which the 98 freshmen at a high school may choose are French and Creative Writing.
How many freshmen enrolled in neither course?
Statement 1: 10 freshmen enrolled in both courses.
Statement 2: 21 freshmen enrolled in each course.
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The question asks for the number of students in the set 
, where 
and 
are the sets of students who took French and Creative Writing, respectively.
From Statement 1 alone, a Venn diagram representing this situation can be filled in as follows:
It is known that 
; subsequently, 
. But no other information is given, so 
, the desired quantity, cannot be calculated.
From Statement 2 alone, a Venn diagram representing this situation can be filled in as follows:
Again, no further information can be computed.
Now assume that both statements are true. Then it follows from Statement 1 that 
, and it follows from Statement 2 that the desired quantity is
.
The question asks for the number of students in the set
From Statement 1 alone, a Venn diagram representing this situation can be filled in as follows:
It is known that
From Statement 2 alone, a Venn diagram representing this situation can be filled in as follows:
Again, no further information can be computed.
Now assume that both statements are true. Then it follows from Statement 1 that
In the above Venn diagram, universal set represents the residents of Wayne. The sets represent the set of all teenagers, those with an even birth month, and males, respectively.
Cary is a resident of Wayne. Is Cary a male?
Statement 1: Cary is a teenager.
Statement 2: Cary does not have an even birth month.
In the above Venn diagram, universal set
Cary is a resident of Wayne. Is Cary a male?
Statement 1: Cary is a teenager.
Statement 2: Cary does not have an even birth month.
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The question asks if Cary is an element of 
.
Assume Statement 1 alone. From the Venn diagram, it can be seen that 
and 
are disjoint sets. Since Cary is an element of 
, he cannot be an element of 
- Cary is not a Male.
Assume Statement 2 alone. From the Venn diagram, it can be seen that 
- that is, if Cary is an element of 
, then she is an element of 
. Restated, if Cary is a Male, then he is a Teenagers. The contrapositive also holds - if Cary is not a teenager - which is given in Statement 2 - then Cary is not a male.
The question asks if Cary is an element of
Assume Statement 1 alone. From the Venn diagram, it can be seen that
Assume Statement 2 alone. From the Venn diagram, it can be seen that
In the above Venn diagram, universal set represents the residents of Belleville. The sets represent the set of all Toastmasters, Elks, and Masons, respectively.
Marty is a resident of Belleville. Is he an Elk?
Statement 1: Marty is neither a Mason nor a Toastmaster.
Statement 2: Marty belongs to exactly one of the three groups.
In the above Venn diagram, universal set
Marty is a resident of Belleville. Is he an Elk?
Statement 1: Marty is neither a Mason nor a Toastmaster.
Statement 2: Marty belongs to exactly one of the three groups.
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The question asks whether or not Marty is an element of 
.
Assume Statement 1 alone. He is an element of 
, represented by the shaded region below:
includes elements that are and are not elements of 
, so it cannot be determined whether or not Marty is in 
.
Assume Statement 2 alone. Then Marty has to be an element of the set represented by the shaded region below:
Since some of the set is in 
and some is not, it cannot be determined whether or not Marty is in 
.
If both statements are known, then, since Marty is in exactly one of the three sets, and he is not a Mason or a Toastmaster, then he must be an Elk.
The question asks whether or not Marty is an element of
Assume Statement 1 alone. He is an element of
Assume Statement 2 alone. Then Marty has to be an element of the set represented by the shaded region below:
Since some of the set is in
If both statements are known, then, since Marty is in exactly one of the three sets, and he is not a Mason or a Toastmaster, then he must be an Elk.
Two of the courses from which the 106 freshmen at Jefferson Academy may choose are American literature and German.
How many freshmen enrolled in both courses?
Statement 1: 19 freshment enrolled in German.
Statement 2: 21 freshmen enrolled in American literature.
Two of the courses from which the 106 freshmen at Jefferson Academy may choose are American literature and German.
How many freshmen enrolled in both courses?
Statement 1: 19 freshment enrolled in German.
Statement 2: 21 freshmen enrolled in American literature.
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Assume both statements are true. If 
is the number of students enrolled in both courses, we can fill in the Venn diagram for the situation with the expressions shown:
No further information is given in the problem, however, so there is no way to calculate 
.
Assume both statements are true. If
No further information is given in the problem, however, so there is no way to calculate
Let universal set 
be the set of all people. Let 
represent Frank.
Let 
be the set of all people who like tea. Let 
be the set of all people who like coffee.
True or false: 
.
Statement 1: Frank likes tea.
Statement 2: Frank likes coffee.
Let universal set
Let
True or false:
Statement 1: Frank likes tea.
Statement 2: Frank likes coffee.
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is the union of the set 
, the set of all people who like coffee, and 
, the complement of the set of all people who like tea - that is, the set of all people who do not like tea. 
if either Frank likes coffee or Frank does not like tea or both.
Statement 1 alone, that Frank likes tea, does not prove or disprove this to be true true; if Frank likes coffee, then this is true, and if he does not like coffee, then this is not true. Statement 2 alone, however, proves this statement true.
Statement 1 alone, that Frank likes tea, does not prove or disprove this to be true true; if Frank likes coffee, then this is true, and if he does not like coffee, then this is not true. Statement 2 alone, however, proves this statement true.
Let universal set be the set of all people. Let 
represent Gina.
Let 
be the set of all people who like cheddar cheese. Let 
be the set of all people who like Swiss cheese.
True or false: 
.
Statement 1: Gina likes cheddar cheese.
Statement 2: Gina does not like Swiss cheese.
Let universal set
Let
True or false:
Statement 1: Gina likes cheddar cheese.
Statement 2: Gina does not like Swiss cheese.
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is the intersection of 
, the complement of the set of people who like cheddar cheese - that is, the set of people who don't like cheddar cheese - and 
, the set of people who like Swiss cheese. For 
to be true, it must hold that Gina doesn't like cheddar cheese and Gina likes Swiss cheese. Either statement alone makes this false.
Let universal set be the set of all people. Let 
represent Holly.
Let 
be the set of people who like The Eagles, 
, the set of people who like Bruce Springsteen, and 
, the set of people who like Crosby, Stills, and Nash.
True or false: Holly likes The Eagles.
Statement 1: 
Statement 2: 
Let universal set
Let
True or false: Holly likes The Eagles.
Statement 1:
Statement 2:
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Assume Statement 1 alone. 
is the intersection of two sets - 
and 
, the complements of, respectively, the set of people who like The Eagles and the set of people who like Bruce Springsteen. Equivalently, this is the intersection of the set of people who do not like The Eagles and the set of people who do not like Bruce Springsteen. Since 
, Holly falls in both sets. Specifically, she does not like The Eagles.
Assume Statement 2 alone. 
is the union of two sets - 
, the set of people who do not like The Eagles and 
, the set of people who like Crosby, Stills, and Nash. If 
, Holly falls in either or both sets, so either she doesn't like The Eagles, or she likes Crosby, Stills, and Nash, or both. It is not resolved whether she likes The Eagles or not.
Assume Statement 1 alone.
Assume Statement 2 alone.
Let universal set be the set of all people. Let 
represent Veronica.
Let 
be the set of people who like Pearl Jam, 
, the set of people who like Nirvana, and 
, the set of people who like Soundgarden.
True or false: Veronica likes at least two of Pearl Jam, Nirvana, and Soundgarden.
Statement 1: 
Statement 2: 
Let universal set
Let
True or false: Veronica likes at least two of Pearl Jam, Nirvana, and Soundgarden.
Statement 1:
Statement 2:
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Assume both statements to be true.
is the union of the sets 
, 
, and 
; Veronica falls in this union, so she falls in at least one of the three sets, so she likes one, two, or all three of Pearl Jam, Nirvana, and Soundgarden.
Similarly, from Statement 2, Veronica falls in at least one of the three sets 
, 
, and 
- the complements of the three sets - meaning that she doesn't like at least one of the three. Therefore, Veronica likes none, one, or two of the artists.
From the two statements together, it follows that Veronica likes one or two of the artists. Without further information, however, it cannot be determined whether she likes two of the artists.
Assume both statements to be true.
Similarly, from Statement 2, Veronica falls in at least one of the three sets
From the two statements together, it follows that Veronica likes one or two of the artists. Without further information, however, it cannot be determined whether she likes two of the artists.
Let universal set be the set of all people. Let 
represent Phillip.
Let 
be the set of people who like Neil Young, 
, the set of people who like Prince, and 
, the set of people who like Marvin Gaye.
True or false: Phillip does not like Neil Young, Prince, or Marvin Gaye.
Statement 1: 
Statement 2: 
Let universal set
Let
True or false: Phillip does not like Neil Young, Prince, or Marvin Gaye.
Statement 1:
Statement 2:
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Assume Statement 1 alone. 
is the intersection of the complements of all three of 
, 
, and 
- the sets of people who do not like Neil Young, people who do not like Prince, and people who do not like Marvin Gaye. Any person who is in this intersection does not like any of these three. 
, so Phillip does not like Neil Young, Prince, or Marvin Gaye.
Assume Statement 2 alone. 
is the union of all three 
, 
, and 
. Any person who is in this union likes any one, two, or three of these musicians. However, 
, which is the complement of this union - the set of people who like none of Neil Young, Prince, or Marvin Gaye. Phil is in this set.
Assume Statement 1 alone.
Assume Statement 2 alone.
Every Bee is a Cee. Every Dee is a Cee. Every Cee is a Fee.
True or false: Bumpy is a Fee.
Statement 1: Bumpy is not a Dee.
Statement 2: Bumpy is not a Bee.
Every Bee is a Cee. Every Dee is a Cee. Every Cee is a Fee.
True or false: Bumpy is a Fee.
Statement 1: Bumpy is not a Dee.
Statement 2: Bumpy is not a Bee.
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and 
.
This can be respresented in a Venn diagram as follows:
Assume both statements to be true. If we let 
represent Bumpy, then from the diagram, it can be seen that if Bumpy is neither a Bee or a Dee, it is possible for Bumpy to be be a Fee or to not be a Fee.
This can be respresented in a Venn diagram as follows:
Assume both statements to be true. If we let
Every Beep is a Deep. Every Deep is a Meep. No Veep is a Deep.
Is Harpy a Veep?
Statement 1: Harpy is a Meep.
Statement 2: Harpy is a Beep.
Every Beep is a Deep. Every Deep is a Meep. No Veep is a Deep.
Is Harpy a Veep?
Statement 1: Harpy is a Meep.
Statement 2: Harpy is a Beep.
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Assume Statement 1 alone. Let 
represent the sets of Beeps, Deeps, Meeps, and Veeps. Then 
and, since all Veeps are not Deeps, 
. This is represented by the Venn diagram below:
Note that the set 
(Meeps) need not fall completely inside the set 
(Veeps). Therefore, Harpy being a Meep does not necessarily make him a Veep or not a Veep.
Assume Statement 2 alone. Every Beep is a Deep, but no Veep is a Deep. If Harpy is a Beep, then he is a Deep. If Harpy were a Veep, then he cannot be a Deep, so, by contradiction, Harpy cannot be a Veep.
Assume Statement 1 alone. Let
Note that the set
Assume Statement 2 alone. Every Beep is a Deep, but no Veep is a Deep. If Harpy is a Beep, then he is a Deep. If Harpy were a Veep, then he cannot be a Deep, so, by contradiction, Harpy cannot be a Veep.
No Fee is a Fi. No Fi is a Fo. No Fo is a Fum.
True or false: Jack is a Fum.
Statement 1: Jack is not a Fo.
Statement 2: Jack is not a Fi.
No Fee is a Fi. No Fi is a Fo. No Fo is a Fum.
True or false: Jack is a Fum.
Statement 1: Jack is not a Fo.
Statement 2: Jack is not a Fi.
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Let 
be the sets of Fe's, Fi's, Fo's and Fum's, respectively. According to the premise of the problem, the sets 
and 
are disjoint (they have no elements in common); the sets 
and 
are disjoint; and the sets 
and 
are disjoint. No other relationships are known. A Venn diagram of these relationships is below:
Assume both statements to be true. The question is whether or not Jack falls in set 
. If Jack is neither a Fi nor a Fo, then Jack falls in neither set 
nor set 
. As evidenced by the two X's in the diagram, it is possible for Jack to fall in set 
or to not fall in set 
. Therefore, the question of whether Jack is a Fum is unresolved.
Let
Assume both statements to be true. The question is whether or not Jack falls in set
Everyone in Jonesville likes exactly three of the following - apples, bananas, cranberries, dates, figs, and guavas. No one in Jonesville who likes apples also likes guavas. No one in Jonesville who likes bananas also likes figs. No one in Jonesville who likes cranberries also likes dates.
Smith lives in Jonesville. True or false: Smith likes figs.
Statement 1: Smith doesn't like bananas.
Statement 2: Smith doesn't like cranberries.
Everyone in Jonesville likes exactly three of the following - apples, bananas, cranberries, dates, figs, and guavas. No one in Jonesville who likes apples also likes guavas. No one in Jonesville who likes bananas also likes figs. No one in Jonesville who likes cranberries also likes dates.
Smith lives in Jonesville. True or false: Smith likes figs.
Statement 1: Smith doesn't like bananas.
Statement 2: Smith doesn't like cranberries.
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Let 
represent the sets of people who like apples, bananas, etc. , respectively. Let 
represent Smith.
Assume Statement 1 alone.
If 
, then 
. Contrapositively, if 
, then 
. Therefore, 
cannot be in both 
and 
; 
is in one, the other, or neither. By similar reasoning, 
is in at most one of 
and 
and at most one of 
and 
.
Suppose 
and 
. Then 
is in three of 
, and either 
falls in both 
and 
or both 
and 
; since both are impossible, it follows that 
or 
, but not both. Therefore, Smith likes exactly one of bananas and figs, but not both. Since, from Statement 1, Smith doesn't like bananas, he must like figs.
Assume Statement 2. By similar reasoning, Smith likes cranberries or dates, but not both. Since he does not like cranberries, he likes dates. However, without further information, it cannot be determined what else he likes or dislikes.
Let
Assume Statement 1 alone.
If
Suppose
Assume Statement 2. By similar reasoning, Smith likes cranberries or dates, but not both. Since he does not like cranberries, he likes dates. However, without further information, it cannot be determined what else he likes or dislikes.
True or false: 
Statement 1: 
is a perfect square.
Statement 2: 
is a multiple of 99.
True or false:
Statement 1:
Statement 2:
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includes all multiples of 2; 
includes all multiples of 3. 
comprises all multiples of either 2 or 3.
Knowing 
is a perfect square is neither necessary nor helpful, as, for example, 
, but 
(as 25 is neither a multiple of 2 nor a multiple of 3).
If you know that 
is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means 
Knowing
If you know that
How many elements are in the set 
?
Statement 1: 
has three more elements than 
.
Statement 2: 
includes exactly four elements not in 
.
How many elements are in the set
Statement 1:
Statement 2:
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Assume both statements are true.
Consider these two cases:
Case1: 
and 
Case 2: 
and 
In both situations, 
has three more elements than 
and 
includes exactly four elements not in 
(1, 2, 3 and 4). However, the number of elements in the union differ in each case - in the first case, 
, and in the second case, 
.
The two statements together do not yield an answer to the question.
Assume both statements are true.
Consider these two cases:
Case1:
Case 2:
In both situations,
The two statements together do not yield an answer to the question.
Define two sets as follows:
where 
and 
are distinct positive odd integers and 
and 
are distinct positive even integers.
How many elements are contained in the set 
?
-
-
Define two sets as follows:
where
How many elements are contained in the set
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Suppose we know 
, but we do not assume the second statement.
If 
and 
, then 
, a four-element set. If If 
and 
, then 
, a three-element set. Therefore, we cannot make a conclusion about the size of 
. A similar argument can be used to show that assuming only the second statement also does not allow a conclusion.
If we know both statements, however, 
, and we can prove that 
has four elements.
The answer is that both statements together are sufficient to answer this question, but neither statement alone is sufficient.
Suppose we know
If
If we know both statements, however,
The answer is that both statements together are sufficient to answer this question, but neither statement alone is sufficient.