Sets - GMAT Quantitative Reasoning
Card 0 of 15
How many subsets does set 
have?
Statement 1: 
has eight elements.
Statement 2: 
is the set of all prime numbers between 1 and 20.
How many subsets does set
Statement 1:
Statement 2:
The number of subsets of any set can be calculated by raising 2 to the power of the number of elements in the set. The first statement gives you that information immediately. The second gives you enough information to find the number of elements, as there are eight primes between 1 and 20: 2, 3, 5, 7, 11, 13, 17, and 19. From either statement alone, you can deduce the answer to be 
.
The number of subsets of any set can be calculated by raising 2 to the power of the number of elements in the set. The first statement gives you that information immediately. The second gives you enough information to find the number of elements, as there are eight primes between 1 and 20: 2, 3, 5, 7, 11, 13, 17, and 19. From either statement alone, you can deduce the answer to be
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Let 
be the set of all of the multiples of 3 between 29 and 50. How many subsets of 
can be formed?
Let
The multiples of 3 between 29 and 50 are 30, 33, 36, 39, 42, 45, and 48 - therefore, 
has seven elements total.
The number of subsets in a set can be calculated by raising 2 to the power of the number of elements. Therefore, the answer to our question is 
.
The multiples of 3 between 29 and 50 are 30, 33, 36, 39, 42, 45, and 48 - therefore,
The number of subsets in a set can be calculated by raising 2 to the power of the number of elements. Therefore, the answer to our question is
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Let 
be the set of all perfect squares and perfect cubes between 1 and 100 inclusive. How many subsets does 
have?
Let
, which is a set of 12 elements. A set this size has 
subsets.
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The senior class has 457 students. An election was held for Senior Class President between three candidates, Anderson, Benson, and Carter.
If every student voted for one of these three, and the student with the most votes was declared the winner, who won the election?
Statement 1: Benson got 251 votes.
Statement 2: Carter got 101 votes.
The senior class has 457 students. An election was held for Senior Class President between three candidates, Anderson, Benson, and Carter.
If every student voted for one of these three, and the student with the most votes was declared the winner, who won the election?
Statement 1: Benson got 251 votes.
Statement 2: Carter got 101 votes.
From Statement 1 alone, it can be immediately deduced that Benson won, since 251 of the 457 votes consititute a majority:
Benson got 54.9% of the vote, so there is no way that Anderson or Carter could have bested Benson.
Statement 2 tells only that Carter did not win, since either Anderson or Benson had to have won at least half, or 
; we cannot, however, tell which one it was without further information.
From Statement 1 alone, it can be immediately deduced that Benson won, since 251 of the 457 votes consititute a majority:
Benson got 54.9% of the vote, so there is no way that Anderson or Carter could have bested Benson.
Statement 2 tells only that Carter did not win, since either Anderson or Benson had to have won at least half, or
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The senior class of Watson High School has 613 students. An election was held for Senior Class President between three candidates, Martindale, Nance, and Osgood.
If every student voted for one of these three, and the student with the most votes was declared the winner, who won the election?
Statement 1: Martindale got 240 votes.
Statement 2: Nance got 244 votes.
The senior class of Watson High School has 613 students. An election was held for Senior Class President between three candidates, Martindale, Nance, and Osgood.
If every student voted for one of these three, and the student with the most votes was declared the winner, who won the election?
Statement 1: Martindale got 240 votes.
Statement 2: Nance got 244 votes.
If we only know Martindale got 240 votes, since 240 is not a majority, we cannot determine the winner with certainty. For example, the following two results are possible:
Martindale: 240, Nance: 0, Osgood 373 - Osgood wins
Martindale 240, Nance 373, Osgood 0 - Nance wins.
A similar argument shows the second statement insufficient as well, since 244 is not a majority.
But the two statements together allow us a complete picture;
Martindale 240, Nance 244, Osgood 129 - Nance wins
If we only know Martindale got 240 votes, since 240 is not a majority, we cannot determine the winner with certainty. For example, the following two results are possible:
Martindale: 240, Nance: 0, Osgood 373 - Osgood wins
Martindale 240, Nance 373, Osgood 0 - Nance wins.
A similar argument shows the second statement insufficient as well, since 244 is not a majority.
But the two statements together allow us a complete picture;
Martindale 240, Nance 244, Osgood 129 - Nance wins
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Two courses open to students in their senior year at Johnson High School are calculus and physics; students may take either or both. Of the 524 seniors enrolled at JHS, are more of them taking calculus or physics?
Statement 1: 139 students are taking neither course.
Statement 2: One-third of the students enrolled in calculus are also enrolled in physics.
Two courses open to students in their senior year at Johnson High School are calculus and physics; students may take either or both. Of the 524 seniors enrolled at JHS, are more of them taking calculus or physics?
Statement 1: 139 students are taking neither course.
Statement 2: One-third of the students enrolled in calculus are also enrolled in physics.
Suppose you know both statements. From the first statement, you can calculate that 
seniors are taking calculus, physics, or both. However, as illustrated by these two examples, you cannot tell which course has more seniors enrolled.
Example 1: 50 seniors are enrolled in both courses.
Then 
seniors are enrolled in calculus; 
seniors are enrolled in physics but not calculus; 
seniors total are enrolled in physics. This means that seniors in physics outnumber seniors in calculus.
Example 2: 100 seniors are enrolled both courses.
Then 
seniors are enrolled in calculus; 
seniors are enrolled in physics but not calculus; 
seniors total are enrolled in physics. This means that seniors in calculus outnumber seniors in physics.
Suppose you know both statements. From the first statement, you can calculate that
Example 1: 50 seniors are enrolled in both courses.
Then
Example 2: 100 seniors are enrolled both courses.
Then
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Given that 
and 
, is it true that positive integer 
?
Statement 1: The last digit of 
is a 0.
Statement 2: The second-to-last digit of 
is 5.
Given that
Statement 1: The last digit of
Statement 2: The second-to-last digit of
and 
are the sets of positive multiples of 4 and 5, respectively. For a number to be in both sets, the number must be divisible by both 4 and 5. This happens if and only if it is also divisible by 
.
The elements of 
are precisely the mulitples of 20:
All of the numbers end in 0 and have 2, 4, 6, 8,or 0 as their second-to-last digit.
Statement 1 does not, by itself, prove or disprove that 
, since there are numbers like 10 and 30 that do not fall in this set. But none of the elements of 
have 5 as their second-to-last digit, so Statement 2 proves 
to be false.
The elements of
All of the numbers end in 0 and have 2, 4, 6, 8,or 0 as their second-to-last digit.
Statement 1 does not, by itself, prove or disprove that
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Define sets 
as follows:
What is 
?
Statement 1: 
Statement 2: 
comprises ten elements, all of which are positive integers.
Define sets
What is
Statement 1:
Statement 2:
is the complement of 
- that is, the set of all elements in the universal set 
that are not in 
. To find 
given 
, we need to know the elements in 
. Statement 1 gives us this information; Statement 2 does not.
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If 
, what is 
?
(1) 
(2) 
If
(1)
(2)
Statement (1) allows us to find B:
B = S + 3 = {4 , 6 , 10} Therefore:
S + B = {1+4 , 1+6 , 1+10 , 3+4 , 3+6 , 3+10 , 7+4 , 7+6 , 7+10}
S + B = {5 , 7 , 9 , 11 , 13 , 17}. SO statement (1) is sufficient to find S+B
Statement (2) does not give us enough information to find B. It could be any set between {4 , 6 , 10} and {1 , 3 , 4 , 6 , 7 , 10} if it includes the same numbers as S. Therefore Statement 2 is not sufficient.
Statement (1) allows us to find B:
B = S + 3 = {4 , 6 , 10} Therefore:
S + B = {1+4 , 1+6 , 1+10 , 3+4 , 3+6 , 3+10 , 7+4 , 7+6 , 7+10}
S + B = {5 , 7 , 9 , 11 , 13 , 17}. SO statement (1) is sufficient to find S+B
Statement (2) does not give us enough information to find B. It could be any set between {4 , 6 , 10} and {1 , 3 , 4 , 6 , 7 , 10} if it includes the same numbers as S. Therefore Statement 2 is not sufficient.
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How many elements are in set 
?
Statement 1: 
has exactly 
subsets.
Statement 2: 
has exactly 
proper subsets.
How many elements are in set
Statement 1:
Statement 2:
A set with 
elements has exactly 
subsets in all, and 
proper subsets (every subset except one - the set itself).
From Statement 1, since 
has 
subsets, it follows that it has 6 elements. From Statement 2, since 
has 63 proper subsets, it has 64 subsets total, and, again, 6 elements. Either statement alone is sufficient.
A set with
From Statement 1, since
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Examine the above Venn diagram, which represents 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: If 
, then 
would be placed in Region III.
Examine the above Venn diagram, which represents the sets of real numbers.
If real number
Statement 1:
Statement 2: If
Statement 1 alone only eliminates Regions I and II (since whole numbers are nonnegative integers); negative numbers can be found in Regions III, IV, and V.
Statement 2 alone states that 
is an integer that as not a whole number - that is, 
is a negative integer. Since 
, as a consequence, 
. 
, the product of a positive integer and a negative integer, is a negative integer, and it would be placed in Region III, which comprises exactly the negative integers.
Statement 1 alone only eliminates Regions I and II (since whole numbers are nonnegative integers); negative numbers can be found in Regions III, IV, and V.
Statement 2 alone states that
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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: If 
, then 
would be placed in Region IV.
Statement 2: If 
, then 
would be placed in Region IV.
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number
Statement 1: If
Statement 2: If
Region IV comprises the rational numbers that are not integers. A number is rational if and only if it can be expressed as the quotient of integers.
From Statement 1 alone, it can be inferred that 
is rational, and that it is not an integer. Since 
, it follows that 
. However, this is not sufficient to narrow it down completely.
For example:
If 
, then 
, a natural number, putting it in Region I.
If 
, then 
, a rational number but not an integer, putting it in Region IV.
From Statement 2 alone, it can be inferred that 
is rational, and that it is not an integer. From 
, it follows that 
. The nonzero rational numbers are closed under division, so 
must be a rational number. However, since 
is not an integer, 
cannot be an integer, since the integers are closed under multiplication. Therefore, Statement 2 alone proves that 
belongs in Region IV.
Region IV comprises the rational numbers that are not integers. A number is rational if and only if it can be expressed as the quotient of integers.
From Statement 1 alone, it can be inferred that
For example:
If
If
From Statement 2 alone, it can be inferred that
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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: If 
, then 
would be placed in Region I.
Statement 2: If 
, then 
would be placed in Region I.
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number
Statement 1: If
Statement 2: If
Region I comprises the natural numbers - 
From Statement 1 alone, 
is a natural number; since 
, it follows that is the difference of a natural number and 7 - that is,
could be in any of three regions - I, II, or III.
Conversely, from Statement 2 alone, 
is the sum of a natural number and 7 - that is,
must be a natural number and it must be in Region I.
Region I comprises the natural numbers -
From Statement 1 alone,
Conversely, from Statement 2 alone,
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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: If 
, then 
would be in Region I.
Statement 2: If 
, then 
would be in Region III.
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number
Statement 1: If
Statement 2: If
Assume Statement 1 alone. It cannot be determined what region 
is in.
For example, suppose 
, which is in Region I (the set of natural numbers, or positive integers). It is possible that 
, putting it in Region I, or 
, putting it in Region III (the set of integers that are not whole numbers - that is, the set of negative integers).
Assume Statement 2 alone. It cannot be determined what region is in.
For example, suppose 
, which is in Region III; then 
, which is also in Region III. But suppose 
; then 
, which, as an irrational number, is in Region V.
Now assume both statements. Then 
has an integer as a square and an integer as a cube. 
must either be an integer or an irrational number. But
, making it the quotient of integers, which is rational. Therefore, 
is an integer. Furthermore, its cube is negative, so 
is negative. The two statements together prove that 
is a negative integer, which belongs in Region III.
Assume Statement 1 alone. It cannot be determined what region
For example, suppose
Assume Statement 2 alone. It cannot be determined what region
For example, suppose
Now assume both statements. Then
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Which, if either, is the greater number: 
or 
?
Statement 1: 
Statement 2: 
Which, if either, is the greater number:
Statement 1:
Statement 2:
Statement 1 alone gives insufficient information. For example, if 
, then:
or 
Since 
, it is unclear which of 
and 
is greater, if either.
Statement 2 gives insufficient information; if 
is positive, 
is negative, and vice versa.
Assume both to be true. The two statements form a system of equations that can be solved using substitution:
Case 1:
Case 2:
This equation has no solution.
Therefore, the only possible solution is 
. Therefore, it can be concluded that 
.
Statement 1 alone gives insufficient information. For example, if
Since
Statement 2 gives insufficient information; if
Assume both to be true. The two statements form a system of equations that can be solved using substitution:
Case 1:
Case 2:
This equation has no solution.
Therefore, the only possible solution is
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