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ISEE Upper Level Quantitative : Sets

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find The Missing Part Of A List

A class has 25 students. If 60% of them are boys, how many students are girls?

Possible Answers:

Correct answer:

Explanation:

If 60% of the students are boys, 40% are girls (100 – 60 = 40). Multiply 25 by 40% (25 * 0.4 = 10); therefore, 10 students are girls.

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Example Question #1 : How To Find The Missing Part Of A List

We can divide the natural numbers into four sets:

$A = \left \{ 1, 5, 9, 13, 17, ... \right \}$

$B= \left \{ 2, 6, 10, 14, 18...\right \}$

$C = \left \{ 3, 7, 11, 15, 19...\right \}$

$D = \left \{ 4, 8, 12, 16, 20...\right \}$

In which of these sets would 197 be a member?

Possible Answers:

It cannot be determined from the information given.

Correct answer:

Explanation:

The sets are divided according to the remainder obtained when each element is divided by 4. 197 divided by 4 yields a remainder of 1; all of the elements of match this description, so $197 \in A$ .

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Example Question #2 : How To Find The Missing Part Of A List

Define the universal set, , as follows:

$U = \left \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$

Also, let

$A = \left \{ 1, 2, 3, 6, 9\right \}$ and $B = \left \{ 3, 5, 7, 9, 10\right \}$

Which of these sets represents the complement of $A \cup B$ ?

Possible Answers:

$\left \{ 3, 4, 8 ,9\right \}$

$\left \{ 1, 2, 3, 5, 6, 7, 9, 10\right \}$

$\left \{ 3,9\right \}$

$\left \{ 4, 8\right \}$

$\left \{ 1, 2, 5, 6, 7, 10\right \}$

Correct answer:

$\left \{ 4, 8\right \}$

Explanation:

$A \cup B$ , the union of and is the set of all elements in , , or both. Merge the two sets to get:

$A \cup B = \left \{ 1, 2, 3, 5, 6, 7, 9, 10\right \}$

The complement of a set is the set of all elements in the universal set that are not in the set. The only elements in not in $A \cup B$ are 4 and 8, so

$\overline{ \left (A \cup B \right )}= \left \{ 4,8\right \}$

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Example Question #1 : How To Find The Missing Part Of A List

Which of the following is an example of two sets and such that $A \cap B = \varnothing$ ?

Possible Answers:

$A = \left \{ \textrm{John, Mary, Fred, Stan, Harry } \right \}$

$B = \left \{ \textrm{Greg, Larry, Phil, Andy, Joanne }\right \}$

$A = \left \{ \Delta , \Gamma , \Theta , \Lambda , \Sigma \right \}$

$B = \left \{ \Omega , \Psi , \Phi , \Pi , \Lambda \right \}$

$A = \left \{ a,v,r, w,s,y\right \}$

$B = \left \{ k, p, r, q,x\right \}$

$A = \left \{ 2, 8, 4, 1, 10\right \}$

$B = \left \{ 5, 9, 7, 2, 0\right \}$

$A = \left \{ \textrm{Lincoln, Washington, Jefferson, Jackson, Madison}\right \}$

$B = \left \{ \textrm{ Monroe, Kennedy, Madison, McKinley , Taft} \right \}$

Correct answer:

$A = \left \{ \textrm{John, Mary, Fred, Stan, Harry } \right \}$

$B = \left \{ \textrm{Greg, Larry, Phil, Andy, Joanne }\right \}$

Explanation:

$\varnothing$ refers to the empty set, the set with no elements; $A \cap B = \varnothing$ if and only if the two sets have no elements in common. In four of these cases, and share an element, which in each of these four choices is underlined:

$A = \left \{ \Delta , \Gamma , \Theta , \underline{\Lambda} , \Sigma \right \}$

$B = \left \{ \Omega , \Psi , \Phi , \Pi , \underline{\Lambda} \right \}$

$A = \left \{ \textrm{Lincoln, Washington, Jefferson, Jackson, \underline{Madison}}\right \}$

$B = \left \{ \textrm{ Monroe, Kennedy, \underline{Madison}, McKinley , Taft} \right \}$

$A = \left \{ a,v,\underline{r}, w,s,y\right \}$

$B = \left \{ k, p, \underline{r}, q,x\right \}$

$A = \left \{ \underline{2}, 8, 4, 1, 10\right \}$

$B = \left \{ 5, 9, 7, \underline{2}, 0\right \}$

$A = \left \{ \textrm{John, Mary, Fred, Stan, Harry } \right \}$ and $B = \left \{ \textrm{Greg, Larry, Phil, Andy, Joanne }\right \}$ do not have an element in common, so this is the right choice.

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Example Question #1 : How To Find The Missing Part Of A List

Define set $A = \left \{ 1, 3, 5, 6, 7, 8, 10 \right \}$ . How could we define set so that $A \cap B = \left \{ 1, 5, 6, 8\right \}$ ?

Possible Answers:

$B = \left \{ 1, 2, 5, 6, 8, 9, 11, 12\right \}$

$B = \left \{ 1,5, 10\right \}$

$B = \left \{ 1, 2, 4, 5, 6, 10, 11, 12\right \}$

$B = \left \{ 1, 2, 5, 6, 9, 12, 13, 15\right \}$

$B = \left \{5, 6,8, 9, 16\right \}$

Correct answer:

$B = \left \{ 1, 2, 5, 6, 8, 9, 11, 12\right \}$

Explanation:

$A \cap B$ is the set of all elements in both and .

We can test each set and determine which elements are shared by that set and :

If $B = \left \{ 1, 2, 5, 6, 9, 12, 13, 15\right \}$ :

then

$A \cap B$

$= \left \{ \underline{1}, 3, \underline{5}, \underline{6}, 7, 8, 10 \right \} \cap \left \{ \underline{1}, 2,\underline{ 5}, \underline{6}, 9, 12, 13, 15\right \}$

$= \left \{ 1,5,6\right \}$

If $B = \left \{ 1, 2, 4, 5, 6, 10, 11, 12\right \}$ :

then

$A \cap B$

$= \left \{ \underline{1}, 3, \underline{5}, \underline{6}, 7, 8, \underline{10} \right \} \cap \left \{ \underline{1}, 2, 4, \underline{5}, \underline{6}, \underline{10}, 11, 12\right \}$

$= \left \{ 1,5,6, 10\right \}$

If $B = \left \{ 1,5, 10\right \}$ :

then

$A \cap B$

$= \left \{ \underline{1}, 3, \underline{5},6, 7, 8, \underline{10} \right \} \cap \left \{ \underline{1}, \underline{5}, \underline{10}\right \}$

$= \left \{ 1,5, 10\right \}$

If $B = \left \{5, 6,8, 9, 16\right \}$ :

then

$A \cap B$

$= \left \{ 1, 3, \underline{5}, \underline{6}, 7, \underline{8}, 10 \right \} \cap\left \{\underline{5}, \underline{6},\underline{8},9, 16\right \}$

$= \left \{ 5,6,8\right \}$

If $B = \left \{ 1, 2, 5, 6, 8, 9, 11, 12\right \}$ :

then

$A \cap B$

$= \left \{ \underline{1}, 3, \underline{5}, \underline{6}, 7, \underline{8}, 10 \right \} \cap \left \{ \underline{1}, 2, \underline{5}, \underline{6}, \underline{8}, 9, 11, 12\right \}$

$= \left \{ 1,5,6, 8\right \}$

This is the correct choice.

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Example Question #2 : How To Find The Missing Part Of A List

A pair of fair dice are rolled. Which is the greater quantity?

(a) The probability that at least one die comes up 5 or 6

(b) $\frac{1}{2}$

Possible Answers:

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given.

(b) is greater

Correct answer:

(a) is greater

Explanation:

For the roll to be unfavorable to the event that at least one of the dice is 5 or 6, both dice would have to be 1, 2, 3, or 4. There are $4 \times 4 = 16$ ways out of 36 that this can happen, so there are 36-16 = 20 ways for either or both of the two dice to be 5 or 6. Since half of 36 is 18, the probability of this event is greater than $\frac{1}{2}$ .

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Example Question #1 : How To Find The Missing Part Of A List

A pair of fair dice are tossed. Which is the greater quantity?

(a) The probability that the product of the two dice will be an even number

(b) $\frac{1}{2}$

Possible Answers:

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

The product of the two dice will be an odd number only if both dice are odd; the rolls favorable to that event are:

$\left \{ (1,1) , (1,3), (1,5), (3,1) , (3,3), (3,5),(5,1) , (5,3), (5,5) \right \}$ ,

or nine out of the thirty-six possible rolls. This makes twenty-seven of the thirty-six equally probable rolls favorable to getting an even product. Since half of 36 is 18, the probability of getting an even product is greater than $\frac{1}{2}$ .

This makes (a) the greater quantity

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Example Question #4 : How To Find The Missing Part Of A List

Which quantity is greater?

(a) The number of even integers that satisfy the inequality 0 < x < 1,000

(b) The number of multiples of 4 that satisfy the inequality 0 < y < 2,000

Possible Answers:

It is impossible to tell from the information given.

(a) is greater

(a) and (b) are equal

(b) is greater

Correct answer:

(a) and (b) are equal

Explanation:

The easiest way to answer this is to try to match each element in the first set to one the second set as follows:

$2 \rightarrow 4$

$4 \rightarrow 8$

$6 \rightarrow 12$

...

$998 \rightarrow 1,996$

In other words, each element in the set in (a) is paired with the element in the set in (b) that is its double. Since there is a one-to-one correspondence, the two sets are of equal aize, and (a) and (b) are equal quantities.

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Example Question #9 : Sets

Which quantity is greater?

(a) The number of even integers that satisfy the inequality $0 < x \leq 1,000$

(b) The number of odd integers that satisfy the inequality $0 < x \leq 1,000$

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

The easiest way to answer this is to try to match each element in the first set to one in the second set as follows:

$2 \rightarrow 1$

$4 \rightarrow3$

$6\rightarrow 5$

...

$1,000 \rightarrow 999$

Since there is a one-to-one correspondence between the elements of the two sets, (a) and (b) are equal.

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Example Question #161 : Data Analysis

A pair of fair dice are tossed. Which is the greater quantity?

(a) The probability that the product of the numbers will be at least .

(b) $\frac {1}{3}$

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

Out of a possible thirty-six rolls, the following result in a product of or greater:

(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6,3), (6, 4), (6, 5), (6, 6)

This is ten equally probable rolls out of thirty-six, resulting in a probability of

$\frac {10}{36} = \frac{5}{18}$ .

Since $\frac {1}{3} = \frac {6} {18} > \frac {5} {18}$ , (b) is the greater quantity.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Sets

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : How To Find The Missing Part Of A List

Example Question #1 : How To Find The Missing Part Of A List

Example Question #2 : How To Find The Missing Part Of A List

Example Question #1 : How To Find The Missing Part Of A List

Example Question #1 : How To Find The Missing Part Of A List

Example Question #2 : How To Find The Missing Part Of A List

Example Question #1 : How To Find The Missing Part Of A List

Example Question #4 : How To Find The Missing Part Of A List

Example Question #9 : Sets

Example Question #161 : Data Analysis

All ISEE Upper Level Quantitative Resources

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