Sets - GMAT Quantitative
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How many elements are in a set from which exactly 768 unique subsets can be formed?
How many elements are in a set from which exactly 768 unique subsets can be formed?
The number of subsets that can be formed from a set with 
elements is 
. However, 
and 
, so there is no integer 
for which 
. Therefore, a set with exactly 768 elements cannot exist.
The number of subsets that can be formed from a set with
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Refer to the above graph, which shows the peak temperature in Smithville for each of seven days in a one-week period.
Between which two consecutuve days did the peak temperature see its greatest decrease?
Refer to the above graph, which shows the peak temperature in Smithville for each of seven days in a one-week period.
Between which two consecutuve days did the peak temperature see its greatest decrease?
You only need to look for the portion of the line with the greatest negative slope, which is that which represents Friday to Saturday.
You only need to look for the portion of the line with the greatest negative slope, which is that which represents Friday to Saturday.
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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.
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 728?
The above represents a Venn diagram. The universal set
Let
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 728?
728 is divisible by 7, but not 3 or 5; therefore, 728 is in 
but not 
or 
. To be in the same region, a number must also be in 
but not 
or 
- that is, divisible by 7 but not 3 or 5.
510 and 595 can be eliminated as multiples of 5 (from the last digit); 777 can be eliminated as a multiple of 3 (digit sum is 21).
Now let's look at the two reminaing choices.
736 is not divisible by 7, since 
.
476 is not divisible by 3 or 5, but it is divisible by 7:
476 is the correct choice.
728 is divisible by 7, but not 3 or 5; therefore, 728 is in
510 and 595 can be eliminated as multiples of 5 (from the last digit); 777 can be eliminated as a multiple of 3 (digit sum is 21).
Now let's look at the two reminaing choices.
736 is not divisible by 7, since
476 is not divisible by 3 or 5, but it is divisible by 7:
476 is the correct choice.
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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 5; let 
be the set of all perfect squares; let 
be the set of perfect cubes.
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 1,225?
The above represents a Venn diagram. The universal set
Let
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 1,225?
1,225 is divisible by 5 (last digit is 5). It is a perfect square, since 
. It is not a perfect cube, however, since 
.
Therefore, 1,225 is an element in 
and 
, but not 
. We are looking for an element in 
and 
, but not 
- that is, a multiple of 5 and a perfect square but not a perfect cube.
By looking at the last digits, we can immediately eliminate 1,764 and 4,356, since neither is a multiple of 5. We can eliminate 15,625, since it s a perfect cube - 
.
Of the two remaining numbers, 3,375 is not a perfect square, since
3,025 is a perfect square: 
3,025 is not a perfect cube: 
3,025 is a multiple of 5, as can be seen from the last digit.
This is the correct choice.
1,225 is divisible by 5 (last digit is 5). It is a perfect square, since
Therefore, 1,225 is an element in
By looking at the last digits, we can immediately eliminate 1,764 and 4,356, since neither is a multiple of 5. We can eliminate 15,625, since it s a perfect cube -
Of the two remaining numbers, 3,375 is not a perfect square, since
3,025 is a perfect square:
3,025 is not a perfect cube:
3,025 is a multiple of 5, as can be seen from the last digit.
This is the correct choice.
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If 
is the set of the multiples of 
between 
and 
and 
is the set of multiples of 
between 
and 
; what is 
?
If
So the intersection of A and B is the numbers that are in both A and B.
Thus 
So the intersection of A and B is the numbers that are in both A and B.
Thus
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Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number 
were to be placed in its correct region in the diagram, which one would it be - I, II, III, IV, or V?
Statement 1: 
is negative.
Statement 2: 
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number
Statement 1:
Statement 2:
If both statements are assumed, then 
- that is, 
. 
could fall in Region IV or V, as is shown in these two examples:
Example 1: 
.
, and 
is rational, so 
would be in Region IV.
Example 2: 
, and 
is irrational, so 
would be in Region V.
The two statements together are insufficient.
If both statements are assumed, then
Example 1:
Example 2:
The two statements together are insufficient.
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Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
and 
are both numbers in Region I; also, 
. In how many of the five regions could the number 
possibly fall?
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
As natural numbers, 
and 
are also rational numbers; since the set of rational numbers is closed under division, and neither 
nor 
is equal to zero (zero not being a natural number), 
is rational and cannot fall in Region V. Regions III (negative integers) and II (zero only) can be eliminated, since both and are positive. This leaves Regions I and IV.
Examples can be produced that would place 
in Region I:
Examples can be produced that would place 
in Region IV (the rational numbers that are not integers):
can fall in either of two different regions.
As natural numbers,
Examples can be produced that would place
Examples can be produced that would place
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Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
and 
are both numbers in Region V, and they may or may not be equal. In how many of the five regions could the number 
possibly fall?
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
The numbers in Region V are the irrational numbers, such as 
and 
.
Since neither number can be the rational number zero, the product of the two cannot be zero, eliminating the possibility that 
. Region II comprises only this number—only 0 is a whole number but not a natural number—so Region II can be eliminated.
Examples can be produced that would place 
in any of the other four regions:
Case 1:
,
placing 
in Region I.
Case 2:
placing 
in Region III (the negative integers).
Case 3:
placing 
in Region IV (the non-integer rational numbers).
Case 4:
placing 
in Region V.
Therefore, 
can fall in any of four different regions.
The numbers in Region V are the irrational numbers, such as
Since neither number can be the rational number zero, the product of the two cannot be zero, eliminating the possibility that
Examples can be produced that would place
Case 1:
placing
Case 2:
placing
Case 3:
placing
Case 4:
placing
Therefore,
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Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
and 
are both numbers in Region II, and they may or may not be equal. In how many of the five regions could the number 
possibly fall?
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
Region II comprises the whole numbers that are not natural numbers; however, there is only one such number, which is 0. Since 
and 
are both numbers in Region II, 
. 
, forcing 
to be in Region II.
Region II comprises the whole numbers that are not natural numbers; however, there is only one such number, which is 0. Since
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Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
is a number in Region I, and 
is a number in Region V. In how many of the five regions could the number 
possibly fall?
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
The key to answering this question is to know that the difference of any two rational numbers is also a rational number.
Suppose 
is rational. Since 
, being a natural number, is also rational, the difference
must be rational.
But it is given that 
is in Region V, making it irrational. This produces a contradiction. Therefore, 
must be irrational, and it can only be in one region, Region V.
The key to answering this question is to know that the difference of any two rational numbers is also a rational number.
Suppose
must be rational.
But it is given that
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What is the fewest number of elements that a set can have in order to have more than 100 subsets?
What is the fewest number of elements that a set can have in order to have more than 100 subsets?
A set with 
elements has exactly 
subsets.
, so a set with 6 elements has 64 subsets.
, so a set with 7 elements has 128 subsets.
The correct response is 7.
A set with
The correct response is 7.
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Given five sets 
, you are given that:
All of the following must be true if 
except:
Given five sets
All of the following must be true if
The subset relation is transitive, so:
and 
together imply that 
;
and 
together imply that 
; and,
and 
together imply that 
.
Since all four of 
are subsets of 
, then any element of any of those four sets is an element of 
. Contrapositively, any nonelement of 
cannot be an element of any of 
.
The subset relation is transitive, so:
Since all four of
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The Fibonacci sequence is the sequence defined as follows:
For all integers 
, 
.
Which of the following expressions is equal to 
?
The Fibonacci sequence is the sequence defined as follows:
For all integers
Which of the following expressions is equal to
Each term after the second is the sum of the preceding two, ao we can relate the 100th term to the 96th and 97th terms as follows:
Each term after the second is the sum of the preceding two, ao we can relate the 100th term to the 96th and 97th terms as follows:
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How many elements are in a set from which exactly 768 unique subsets can be formed?
How many elements are in a set from which exactly 768 unique subsets can be formed?
The number of subsets that can be formed from a set with elements is . However, and , so there is no integer for which . Therefore, a set with exactly 768 elements cannot exist.
The number of subsets that can be formed from a set with
Compare your answer with the correct one above
Refer to the above graph, which shows the peak temperature in Smithville for each of seven days in a one-week period.
Between which two consecutuve days did the peak temperature see its greatest decrease?
Refer to the above graph, which shows the peak temperature in Smithville for each of seven days in a one-week period.
Between which two consecutuve days did the peak temperature see its greatest decrease?
You only need to look for the portion of the line with the greatest negative slope, which is that which represents Friday to Saturday.
You only need to look for the portion of the line with the greatest negative slope, which is that which represents Friday to Saturday.
Compare your answer with the correct one above
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.
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 728?
The above represents a Venn diagram. The universal set
Let
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 728?
728 is divisible by 7, but not 3 or 5; therefore, 728 is in but not or . To be in the same region, a number must also be in but not or - that is, divisible by 7 but not 3 or 5.
510 and 595 can be eliminated as multiples of 5 (from the last digit); 777 can be eliminated as a multiple of 3 (digit sum is 21).
Now let's look at the two reminaing choices.
736 is not divisible by 7, since .
476 is not divisible by 3 or 5, but it is divisible by 7:
476 is the correct choice.
728 is divisible by 7, but not 3 or 5; therefore, 728 is in
510 and 595 can be eliminated as multiples of 5 (from the last digit); 777 can be eliminated as a multiple of 3 (digit sum is 21).
Now let's look at the two reminaing choices.
736 is not divisible by 7, since
476 is not divisible by 3 or 5, but it is divisible by 7:
476 is the correct choice.
Compare your answer with the correct one above
The above represents a Venn diagram. The universal set is the set of all positive integers.
Let be the set of all multiples of 5; let be the set of all perfect squares; let be the set of perfect cubes.
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 1,225?
The above represents a Venn diagram. The universal set
Let
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 1,225?
1,225 is divisible by 5 (last digit is 5). It is a perfect square, since . It is not a perfect cube, however, since .
Therefore, 1,225 is an element in and , but not . We are looking for an element in and , but not - that is, a multiple of 5 and a perfect square but not a perfect cube.
By looking at the last digits, we can immediately eliminate 1,764 and 4,356, since neither is a multiple of 5. We can eliminate 15,625, since it s a perfect cube - .
Of the two remaining numbers, 3,375 is not a perfect square, since
3,025 is a perfect square:
3,025 is not a perfect cube:
3,025 is a multiple of 5, as can be seen from the last digit.
This is the correct choice.
1,225 is divisible by 5 (last digit is 5). It is a perfect square, since
Therefore, 1,225 is an element in
By looking at the last digits, we can immediately eliminate 1,764 and 4,356, since neither is a multiple of 5. We can eliminate 15,625, since it s a perfect cube -
Of the two remaining numbers, 3,375 is not a perfect square, since
3,025 is a perfect square:
3,025 is not a perfect cube:
3,025 is a multiple of 5, as can be seen from the last digit.
This is the correct choice.
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If is the set of the multiples of between and and is the set of multiples of between and ; what is ?
If
So the intersection of A and B is the numbers that are in both A and B.
Thus
So the intersection of A and B is the numbers that are in both A and B.
Thus
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Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number were to be placed in its correct region in the diagram, which one would it be - I, II, III, IV, or V?
Statement 1: is negative.
Statement 2:
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number
Statement 1:
Statement 2:
If both statements are assumed, then - that is, . could fall in Region IV or V, as is shown in these two examples:
Example 1: .
, and is rational, so would be in Region IV.
Example 2:
, and is irrational, so would be in Region V.
The two statements together are insufficient.
If both statements are assumed, then
Example 1:
Example 2:
The two statements together are insufficient.
Compare your answer with the correct one above
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
and are both numbers in Region I; also, . In how many of the five regions could the number possibly fall?
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
As natural numbers, and are also rational numbers; since the set of rational numbers is closed under division, and neither nor is equal to zero (zero not being a natural number), is rational and cannot fall in Region V. Regions III (negative integers) and II (zero only) can be eliminated, since both and are positive. This leaves Regions I and IV.
Examples can be produced that would place in Region I:
Examples can be produced that would place in Region IV (the rational numbers that are not integers):
can fall in either of two different regions.
As natural numbers,
Examples can be produced that would place
Examples can be produced that would place
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