Sets - GMAT Quantitative
Card 1 of 168
A fair coin is flipped successively until heads are observed on 2 successive flips. Let x denote the number of coin flips required. What is the sample space of x?
A fair coin is flipped successively until heads are observed on 2 successive flips. Let x denote the number of coin flips required. What is the sample space of x?
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We need to flip a coin until we get two heads in a row. The smallest number of possible flips is 2, which would occur if our first two flips are both heads. This eliminates three of our answer choices, because we know the sample space must start at 2.
This leaves us with {x : x = 2, 3, 4 . . .} and {x : x = 2, 3, 4, 5, 6}. Let's think about {x : x = 2, 3, 4, 5, 6}. What if I flip a coin 6 times and get 6 tails? Then I have to keep flipping beyond 6 flips until I get two heads in a row; therefore the answer must be {x : x = 2, 3, 4 . . .}, because we don't have an upper limit on the number of flips it will take to produce two successive heads.
We need to flip a coin until we get two heads in a row. The smallest number of possible flips is 2, which would occur if our first two flips are both heads. This eliminates three of our answer choices, because we know the sample space must start at 2.
This leaves us with {x : x = 2, 3, 4 . . .} and {x : x = 2, 3, 4, 5, 6}. Let's think about {x : x = 2, 3, 4, 5, 6}. What if I flip a coin 6 times and get 6 tails? Then I have to keep flipping beyond 6 flips until I get two heads in a row; therefore the answer must be {x : x = 2, 3, 4 . . .}, because we don't have an upper limit on the number of flips it will take to produce two successive heads.
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Mark will hire 5 of the 8 job applicants he interviews. In how many different ways can he do this?
Mark will hire 5 of the 8 job applicants he interviews. In how many different ways can he do this?
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Since order doesn't matter here, set this up as a combination:
Since order doesn't matter here, set this up as a combination:
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Out of 2,319 employees of a company, 1,299 speak Spanish and 1,122 speak French; 517 speak neither. How many speak both?
Out of 2,319 employees of a company, 1,299 speak Spanish and 1,122 speak French; 517 speak neither. How many speak both?
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Let 
and 
be the sets of employees that speak Spanish and French, respectively. Then we know that
and 
Since 517 employees speak neither, 
employees speak either Spanish, French, or both.
We are looking for 
, which can be found by using the formula:
Let
Since 517 employees speak neither,
We are looking for
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Refer to the Venn diagram. Let universal set 
be the set of all natural numbers, 
.
Let 
be the set of all multiples of 
; let 
be the set of all perfect squares; let 
be the set of all perfect cubes. Which region of the Venn diagram contains the number 
?
Refer to the Venn diagram. Let universal set
Let
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, making 1,728 a multiple of 3, and thus, an element of 
.
1,728 is not a perfect square; 
. Thus, 1,728 is not an element of 
.
1,728 is a perfect cube: 
. Thus, 1,728 is an element of 
.
, which is represented by the region inside circles 
and 
and outside 
. This is region 
.
1,728 is not a perfect square;
1,728 is a perfect cube:
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Which of these Venn diagrams represents the set 
?
Which of these Venn diagrams represents the set
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is the set of elements that fall either in 
or the complement of 
, or both - that is, either in 
, outside of 
, or both. This union is intersected with the complement of 
, meaning that only the elements of the union that also fall outside of 
are considered.
"Color" in all of 
and everything outside of 
- but then, uncolor everything inside 
. That makes the correct choice:
"Color" in all of
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What is the median of the following number set?
What is the median of the following number set?
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In order to find the median, the set needs to be written in numerical order:
Since 
and 
are both the middle numbers, taking their average will give the median of the set.
In order to find the median, the set needs to be written in numerical order:
Since
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The above represents a Venn diagram. The universal set 
is the set of all positive integers.
Let 
be the set of all multiples of 3; let 
be the set of all multiples of 5; let 
be the set of all multiples of 7. Which of the five marked regions would include the number 525?
The above represents a Venn diagram. The universal set
Let
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525 is a multiple of all three of the integers 3, 5, and 7:
Therefore, 525 is an element of each of sets 
, and, subsequently, falls into region 
, which represents 
.
525 is a multiple of all three of the integers 3, 5, and 7:
Therefore, 525 is an element of each of sets
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In a group of 30 freshman students, 10 are taking Pre-calculus, 15 are taking Biology, and 10 students are taking Algebra. 5 Students are taking both Algebra and Biology, and 7 students are taking both Biology and Pre-calculus. There is no student taking both Algebra and Pre-Calculus. If none of the students take the three classes together, how many of the students don't take any of the three classes?
In a group of 30 freshman students, 10 are taking Pre-calculus, 15 are taking Biology, and 10 students are taking Algebra. 5 Students are taking both Algebra and Biology, and 7 students are taking both Biology and Pre-calculus. There is no student taking both Algebra and Pre-Calculus. If none of the students take the three classes together, how many of the students don't take any of the three classes?
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Let 
be the number of students who don't take any of the three classes.
Let
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Set B contains all prime numbers. Set C contains all even numbers. How many numbers are common to both sets?
Set B contains all prime numbers. Set C contains all even numbers. How many numbers are common to both sets?
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Prime numbers are numbers with no other factors than themselves and one. Two is the first prime number and the only even prime number. Other examples are 5, 7, 11, etc.
Even numbers are numbers divisible by 2. Set C includes all numbers ending in 0, 2, 4, 6, or 8.
Thus, there is one number common to both sets: 2.
Prime numbers are numbers with no other factors than themselves and one. Two is the first prime number and the only even prime number. Other examples are 5, 7, 11, etc.
Even numbers are numbers divisible by 2. Set C includes all numbers ending in 0, 2, 4, 6, or 8.
Thus, there is one number common to both sets: 2.
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If universal set 
refers to the set of seniors at Washington High School, 
is the set of seniors enrolled in physics, 
is the set of seniors enrolled in calculus, and 
is the set of seniors enrolled in French IV, then the above Venn diagram reflects all of the following except:
If universal set
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The sets 
and 
do not intersect, so no senior is enrolled in both French IV and physics; the sets 
and 
do not intersect, so no senior is enrolled in both French IV and calculus.
, so every senior enrolled in calculus is also enrolled in physics; contrapositively, every senior not enrolled in physics is also not enrolled in calculus.
The correct choice is the remaining statement - every senior enrolled in physics is also enrolled in calculus - since 
is not a subset of 
.
The sets
The correct choice is the remaining statement - every senior enrolled in physics is also enrolled in calculus - since
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Choose the statement that is the logical opposite of:
"John is a Toastmaster but not an Elk."
Choose the statement that is the logical opposite of:
"John is a Toastmaster but not an Elk."
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Let 
and 
be the set of all Toastmasters and Elks, respectively, and let 
be the set of all people. 
and 
, so the set to which John belongs is the shaded set in this Venn diagram:
the logical opposite of this is that John belongs to the shaded set in the diagram:
A way of saying this is 
or 
, or, equivalently, if 
, then 
.
In plain English, if John is not an Elk, then John is not a Toastmaster.
Let
the logical opposite of this is that John belongs to the shaded set in the diagram:
A way of saying this is
In plain English, if John is not an Elk, then John is not a Toastmaster.
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Every Moog is a Noog. Every Noog is a Poog. Every Poog is a Roog.
True or False: Stumpy is a Moog.
Statement 1: Stumpy is not a Poog.
Statement 2: Stumpy is not a Roog.
Every Moog is a Noog. Every Noog is a Poog. Every Poog is a Roog.
True or False: Stumpy is a Moog.
Statement 1: Stumpy is not a Poog.
Statement 2: Stumpy is not a Roog.
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Let 
be the sets of Moogs, Noogs, Poogs, and Roogs, respectively, From the premise of the problem, we have that
.
It follows by transitivity that 
and 
.
Assume Statement 1 alone. If Stumpy is a Moog, then
.
, makng Stumpy a Poog. But since Stumpy is NOT a Poog, then it follows, contrapositively, that Stumpy is not a Moog.
Assume Statement 2 alone. If Stmpy is a Moog, then
,
making Stumpy a Roog. Since Stumpy is NOT a Roog, then it follows, contrapositively, that Stumpy is not a Moog.
Let
It follows by transitivity that
Assume Statement 1 alone. If Stumpy is a Moog, then
, makng Stumpy a Poog. But since Stumpy is NOT a Poog, then it follows, contrapositively, that Stumpy is not a Moog.
Assume Statement 2 alone. If Stmpy is a Moog, then
making Stumpy a Roog. Since Stumpy is NOT a Roog, then it follows, contrapositively, that Stumpy is not a Moog.
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Let 
be the universal set of all people. Let 
be the set of all people who like catfish. Let 
be the set of all people who like flounder.
Let 
represent Wendy. Which of these statements states the opposite of "Wendy likes neither catfish nor flounder" in set notation?
Let
Let
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In set theory, "Wendy likes neither catfish nor flounder" can be stated as
.
The opposite of this is that 
is in the complement of this set:
.
By DeMorgan's law, this is equivalent to saying
, or
.
In set theory, "Wendy likes neither catfish nor flounder" can be stated as
The opposite of this is that
By DeMorgan's law, this is equivalent to saying
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Let universal set 
be the set of all people. Let 
be the set of people who like The Rolling Stones. Let 
be the set of people who like The Who. Let 
be the set of people who like Lynyrd Skynyrd.
Let 
stand for Nancy.
True or false: 
Statement 1: Nancy does not like The Who.
Statement 2: Nancy likes Lynyrd Skynyrd.
Let universal set
Let
True or false:
Statement 1: Nancy does not like The Who.
Statement 2: Nancy likes Lynyrd Skynyrd.
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If , then 
falls in the intersection of two sets. One is 
, the union of 
, the set of people who like The Rolling Stones, and the complement of 
, the set of people not in 
- the set of all people who do not like The Who. The other set is the complement of 
, the set of all people not in 
- the set of people who don't like Lynyrd Skynyrd.
Therefore, if , both of the following must happen: Nancy must like The Rolling Stones or not like The Who, or both, and Nancy must not like Lynyrd Skynyrd.
From Statement 1 alone, Nancy does not like The Who, but it is not clear who else she likes. She is in 
, but it is not clear whether she is in 
the set of people who don't like Lynyrd Skynyrd, so the question of whether she falls in 
is open.
From Statement alone, since Nancy likes Lynyrd Skynyrd, she cannot be in 
. This makes false.
If
Therefore, if
From Statement 1 alone, Nancy does not like The Who, but it is not clear who else she likes. She is in
From Statement alone, since Nancy likes Lynyrd Skynyrd, she cannot be in
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Let be the universal set of all people. Let 
be the set of all people who can swim. Let 
be the set of all people who can ride a bicycle.
Let 
represent Kim. Which of the following English-language sentences is the opposite of 
?
Let
Let
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The opposite of the statement 
is the statement
,
which, by DeMorgan's Law, is equivalent to
or
This means that Kim is in the intersection of 
and 
, or, equivalently, she is in both 
, the set of people who can swim, and 
, the complement of the set of people who can ride a bicycle. Therefore, Kim can swim, but she cannot ride a bicycle.
The opposite of the statement
which, by DeMorgan's Law, is equivalent to
or
This means that Kim is in the intersection of
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Let be the universal set of all people. Let be the set of all people who like cola. Let be the set of all people who like milk.
Let 
represent Victor. Which of the following English-language sentences is the opposite of 
?
Let
Let
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The opposite of the statement 
is the statement
,
which, by DeMorgan's Law, is equivalent to
, or
Victor belongs to either or both of 
, the set of all people who like cola, and 
, the set of all people who like milk. Therefore, Victor likes cola, milk, or both.
The opposite of the statement
which, by DeMorgan's Law, is equivalent to
Victor belongs to either or both of
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Let universal set 
represent the set of all restaurants in the city of Bancroft. Let 
represent the set of all restaurants in Bancroft that accept the America's Best credit card. Let 
represent the set of all restaurants in Bancroft that accept the Bank Zero credit card. Let 
represent the set of all restaurants that accept the Chargitall credit card.
Of the following five restaurants in Bancroft:
Ally's accepts America's Best and Bank Zero, but not Chargitall.
Benny's accepts Chargitall, but not American's Best or Bank Zero.
Carleton's accepts America's Best, but neither Bank Zero nor Chargitall.
Danny's accepts Bank Zero, but neither America's Best nor Chargitall.
Eddie's does not accept any of the three cards.
Which of the following restaurants would fall in the set 
?
Let universal set
Of the following five restaurants in Bancroft:
Ally's accepts America's Best and Bank Zero, but not Chargitall.
Benny's accepts Chargitall, but not American's Best or Bank Zero.
Carleton's accepts America's Best, but neither Bank Zero nor Chargitall.
Danny's accepts Bank Zero, but neither America's Best nor Chargitall.
Eddie's does not accept any of the three cards.
Which of the following restaurants would fall in the set
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To fall in the set 
, a restaurant would have to be in the union of three sets - that is, any one or more. The three sets are:
, the set of all restaurants in Bancroft that accept the America's Best credit card;
, the set of all restaurants in Bancroft that accept the Bank Zero credit card; and
, the complement of the set of all restaurants in Bancroft that accept the Chargitall credit card, or, equivalently, set of all restaurants in Bancroft that do not accept the Chargitall credit card.
The complement of this set, 
, is the set of all restaurants that belong to none of 
, so the correct choice must be a restaurant that belongs to 
and 
. The restaurant must accept Chargitall, and must not accept the other two cards.
The only one of the five restaurants fitting this description is Benny's.
To fall in the set
The complement of this set,
The only one of the five restaurants fitting this description is Benny's.
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Let be the universal set of all people. Let 
be the set of all people who like Frank Sinatra. Let 
be the set of all people who like The Beatles. Let 
be the set of all people who like Elvis Presley. Let 
stand for Tommy.
Let 
. Which of the following must be false?
Let
Let
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We can best answer this question using a Venn diagram. We want to examine the union of two sets:
, the intersection of 
and 
, which is the region the two circles have in common, and
, the complement 
, which is the region outside the circle representing this set.
Since we are examing the union, we want to shade in everything in one or both of these two sets. This is shown below:
If Tommy likes Frank Sinatra, The Beatles, and Elvis Presley, 
falls in all three circles, placing him in Region (i).
If Tommy does not like Frank Sinatra, The Beatles, or Elvis Presley, 
falls in none of the three circles, placing him in Region (ii).
If Tommy likes Frank Sinatra, but neither The Beatles nor Elvis Presley, 
falls in circle A, but neither B nor C, placing him in Region (iii).
If Tommy likes Frank Sinatra and The Beatles, but not Elvis Presley, 
falls in circles A and B, but not C, placing him in Region (iv).
If Tommy likes Frank Sinatra and Elvis Presley, but not The Beatles, 
falls in circles A and C, but not B, placing him in Region (v).
In all of the first four cases, 
is a true statement, since the region is part of the grayed area. In the fifth case, 
is false. Therefore, the only impossible statement among the choices is "Tommy likes Frank Sinatra and Elvis Presley, but not The Beatles."
We can best answer this question using a Venn diagram. We want to examine the union of two sets:
Since we are examing the union, we want to shade in everything in one or both of these two sets. This is shown below:
If Tommy likes Frank Sinatra, The Beatles, and Elvis Presley,
If Tommy does not like Frank Sinatra, The Beatles, or Elvis Presley,
If Tommy likes Frank Sinatra, but neither The Beatles nor Elvis Presley,
If Tommy likes Frank Sinatra and The Beatles, but not Elvis Presley,
If Tommy likes Frank Sinatra and Elvis Presley, but not The Beatles,
In all of the first four cases,
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Let be the universal set of all people. Let be the set of all people who like Band A. Let be the set of all people who like Band B. Let be the set of all people who like Band C. Let 
stand for Julianna.
Let 
. Which of the following could be true?
Let
Let
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, which is the intersection of 
and 
. It follows that 
and 
.
, so it follows that Julianna likes Band A. The three choices that state that she does not can be eliminated.
. This is the union of 
, the complement of 
- that is, the set of people not in 
- and 
. It follows that either Julianna does not like Band B, does like Band C, or both. Therefore, it is not true that she likes Band B and does not like Band C. This can be eliminated.
The only possible choice is that Julianna likes Band A and Band C, but not Band B.
The only possible choice is that Julianna likes Band A and Band C, but not Band B.
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Let be the universal set of all people. Let 
be the set of all people who are dancers and 
be the set of all people that are singers.
Let 
represent Jeremy. Which of these statements states the opposite of "Jeremy is not a dancer, but he is a singer" in set notation?
Let
Let
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In set notation, "Jeremy is not a dancer, but he is a singer" can be stated as
That is, Jeremy falls in the intersection of the complement of the set of dancers 
and the set of singers 
.
The opposite of this is that 
is in the complement of this set:
By DeMorgan's law, this is equivalent to saying
, or
.
In set notation, "Jeremy is not a dancer, but he is a singer" can be stated as
That is, Jeremy falls in the intersection of the complement of the set of dancers
The opposite of this is that
By DeMorgan's law, this is equivalent to saying
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