Sets - GMAT Quantitative Reasoning

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Question

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?

Answer

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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Question

Which set is NOT equal to the other sets?

Answer

Order and repetition do NOT change a set. Therefore, the set we want to describe contains the numbers 1, 3, and 4. The only set that doesn't contain all 3 of these numbers is $\left { 1,3,1,1 \right }$ , so it is the set that does not equal the rest of the sets.

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Question

Given the sets A = {2, 3, 4, 5} and B = {3, 5, 7}, what is $A \bigcup B$ ?

Answer

We are looking for the union of the sets. That means we want the elements of A OR B.

So $A \bigcup B$ = {2, 3, 4, 5, 7}.

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Question

Given the set = {2, 3, 4, 5}, what is the value of ?

Answer

We need to add 3 to every element in .

Then:

$3+A=\left { 3+2,3+3,3+4,3+5 \right }=\left { 5,6,7,8 \right }.$

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Question

There exists two sets and . = {1, 4} and = {3, 4, 6}. What is ?

Answer

Add each element of to each element of .

= {1 + 3, 1 + 4, 1 + 6, 4 + 3, 4 + 4, 4 + 6} = {4, 5, 7, 8, 10}

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Question

How many functions map from $X=\left { a,b \right }$ to $Y=\left { 1,2,3 \right }$ ?

Answer

There are three choices for (1, 2, and 3), and similarly there are three choices for (also 1, 2, and 3). Together there are $3\cdot 3=9$ possible functions from to . Remember to multiply, NOT add.

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Question

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?

Answer

$728 \div 3 = 242\textrm{ R }2$

$728 \div 5 = 145\textrm{ R }3$

$728 \div 7 = 104$

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 $736 \div 7 = 105 \textrm{ R }1$ .

476 is not divisible by 3 or 5, but it is divisible by 7:

$476 \div 3 = 158\textrm{ R }2$

$476 \div 5 = 95\textrm{ R }1$

$476 \div 7 = 68$

476 is the correct choice.

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Question

How many elements are in a set from which exactly 768 unique subsets can be formed?

Answer

The number of subsets that can be formed from a set with elements is $2^{N}$ . However, $2^{9} = 512$ and $2^{10} = 1024$ , so there is no integer for which $2^{N} = 768$ . Therefore, a set with exactly 768 elements cannot exist.

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Question

Let the univeraal set be the set of all positive integers.

Define the sets

$A = \left { 1,6,11,16,21,26,...\right}$ ,

$B = \left {1, 5, 9, 13,17,21,25,...\right }$ ,

$C = \left{1,4,7,10,13,16,19,...\right }$ .

If the elements in $A \cap B \cap C$ were ordered in ascending order, what would be the fourth element?

Answer

are the sets of all positive integers that are one greater than a multiple of five, four, and three, respectively. Therefore, for a number to be in all three sets, and subsequently, $A \cap B \cap C$ , the number has to be one greater than a number that is a multiple of five, four, and three. Since , the number has to be one greater than a multiple of 60. The first four numbers that fit this description are 1, 61, 121, and 181, the last of which is the correct choice.

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Question

A six-sided die is rolled, and a coin is flipped. If the coin comes up heads, the roll is considered to be the number that appears face up on the die; if the coin comes up tails, the outcome is considered to be half that number, with any fraction simply thrown out. What is the sample space of the experiment?

Answer

If heads comes up on the coin, the number on the die is recorded. This can be any element of the set $\left { 1,2,3,4,5,6\right }$ .

If tails comes up on the coin, half the number on the die is recorded, and any fraction is thrown out. Half the elements of the set $\left { 1,2,3,4,5,6\right }$ comprise the set $\left { \frac{1}{2}, 1, 1\frac{1}{2}, 2,2 \frac{1}{2}, 3\right }$ , but since we are throwing out the fractional parts, this becomes the set $\left { 0, 1,2,3\right }$ .

The sample space is the union of these two sets, which is $\left { 0,1,2,3,4,5,6\right }$ .

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Question

A six-sided die is rolled, and a coin is flipped. If the coin comes up heads, the roll is considered to be the number that appears face up on the die; if the coin comes up tails, the outcome is considered to be twice that number. What is the sample space of the experiment?

Answer

If heads comes up on the coin, the number on the die is recorded. This can be any element of the set $\left { 1,2,3,4,5,6\right }$ .

If tails comes up on the coin, twice the number on the die is recorded. This can be twice any element of the set $\left { 1,2,3,4,5,6\right }$ - that is, any element of the set $\left { 2,4,6,8,10,12\right }$ .

The sample space is the union of these two sets:

$\left { 1,2,3,4,5,6 ,8,10,12\right }$

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Question

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?

Answer

1,225 is divisible by 5 (last digit is 5). It is a perfect square, since $1,225 = 35 ^{2}$ . It is not a perfect cube, however, since $10 ^{3} = 1,000 < 1,225 < 1,331 = 11 ^{3}$ .

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 - $15,625 = 25 ^{3}$ .

Of the two remaining numbers, 3,375 is not a perfect square, since

$58 ^{2} = 3,364 < 3,375 < 3,481 = 59^{2}$

3,025 is a perfect square: $35 ^{2} = 3,025$

3,025 is not a perfect cube: $14 ^{3} = 2,744 < 3,025 < 15^{3}= 3,375$

3,025 is a multiple of 5, as can be seen from the last digit.

This is the correct choice.

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Question

Let be the set ${\Gamma ,\Delta ,\Lambda ,\Phi }$ , and be the set ${\Pi ,\Lambda ,\Delta ,\Theta ,\Gamma ,\Phi ,\Omega , C^2}$ .

What are the elements in the set $A \cap B$ ?

Answer

$A \cap B$ is the set of all elements that are in both and . So in this case the elements that are the same for both sets are $\Delta,\Lambda,\Gamma,\Phi$ , Note that the order that you present the elements in your set doesn't matter. What matters is that you don't exclude any necessary elements, or add any that don't belong.

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Question

If is the set of the multiples of between and and is the set of multiples of between and ; what is $A \bigcap B$ ?

Answer

So the intersection of A and B is the numbers that are in both A and B.

Thus $A \bigcap B= [6 , 12]$

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Question

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:

Answer

If both statements are assumed, then - that is, $n \in (-1, 0)$ . could fall in Region IV or V, as is shown in these two examples:

Example 1: $n= -\frac{1}{2}$ .

, and is rational, so would be in Region IV.

Example 2: $n= -\frac{\pi}{4}$

, and is irrational, so would be in Region V.

The two statements together are insufficient.

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Question

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 $m \div n$ possibly fall?

Answer

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), $m \div n$ 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 $m \div n$ in Region I:

$m = 2, n = 1 \Rightarrow m \div n = 2$

Examples can be produced that would place $m \div n$ in Region IV (the rational numbers that are not integers):

$m = 1, n = 2 \Rightarrow m \div n = \frac{1}{2}$

$m \div n$ can fall in either of two different regions.

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Question

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?

Answer

The numbers in Region I are exactly the natural numbers $\left { 1, 2, 3, 4, 5,...\right }$ . All natural numbers are integers, and the integers are closed under subtraction, so cannot fall in Region IV or Region V. Also, since it is given in the problem that , it follows that , so the difference cannot be in Region II (the only whole number that is not a natural number is ).

It is possible for to be in Region I. Example:

$m = 2, n = 1 \Rightarrow m - n = 1$

It is possible for to be in Region III (the integers that are not whole numbers, or the negative integers). Example:

$m = 1, n = 2 \Rightarrow m - n= -1$

can fall in either of two different regions.

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Question

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?

Answer

The numbers in Region V are the irrational numbers, such as $\sqrt{2}$ and $\pi$ .

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:

$m = \sqrt{2}, n = \sqrt{2} \Rightarrow mn = \sqrt{2}\cdot \sqrt{2} = 2$ ,

placing in Region I.

Case 2:

$m =- \sqrt{2}, n = \sqrt{2} \Rightarrow mn =- \sqrt{2}\cdot \sqrt{2} =- 2$

placing in Region III (the negative integers).

Case 3:

$m = \sqrt{\frac{2}{3}}, n = \sqrt{\frac{2}{3}} \Rightarrow mn = \sqrt{\frac{2}{3}}\cdot \sqrt{\frac{2}{3}} = \frac{2}{3}$

placing in Region IV (the non-integer rational numbers).

Case 4:

$m = \sqrt{3}, n = \sqrt{2} \Rightarrow mn = \sqrt{3}\cdot \sqrt{2} = \sqrt{6}$

placing in Region V.

Therefore, can fall in any of four different regions.

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Question

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?

Answer

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.

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Question

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?

Answer

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.

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