# ISEE Upper Level Quantitative : Venn Diagrams

## Example Questions

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### Example Question #1 : Venn Diagrams

Let the universal set  be the set of all positive integers. Define:

Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?

None of the other choices is correct.

None of the other choices is correct.

Explanation:

The grayed portion of the Venn diagram corresponds to those integers which are not in any of , or . Therefore, we eliminate any choices that are in any of the three sets.

is the set of integers which end in 1 or 6; we can eliminate 166 and 176 immediately.

is the set of perfect square integers; we can eliminate 144, since .

is the set of integers which, when divided by 4, yields remainder 2. Since  , we can eliminate 154.

All four choices have been eliminated.

### Example Question #1 : How To Use A Venn Diagram

In the above Venn diagram, the universal set is defined as  . Each of the eight letters is placed in its correct region.

What is ?

Explanation:

is the union of sets  and  - that is, the set of all elements of  that are elements of either  or . We want all of the letters that fall in either circle, which from the diagram can be seen to be all of the letters except . Therefore,

### Example Question #1 : Data Analysis And Probability

In a school of  students,  students take Greek,  take Old English, and  take neither. How many take both?

Explanation:

Based on the information given, you can construct the following Venn Diagram:

In order to find the overlap, you need to find out how many are in the circles together. This is easy. Subtract: . Now, since the overlap represents a duplication, you need to subtract out one of those duplicate values. Let's call that ; therefore, we know that:

Solving for , you get:

### Example Question #1 : Venn Diagrams

In a group of  people,  have a laptop and  have a tablet. Of those people who have a laptop or a tablet,  have both. How many people in the total group have neither a laptop nor a tablet?

Explanation:

Based on the information given, you can draw the following Venn Diagram:

To solve this, remember that the total number of values in the two circles is:

(We must do this because of the overlap.  You need to subtract out one instance of that overlap.)

If we assign the value  for the unknown region, we know:

### Example Question #1 : Venn Diagrams

In a group of plants,  are green,  have large leaves, and  are both green and have large leaves. How many plants are green without large leaves?

Explanation:

Based on the information, you can draw the following Venn Diagram:

It is very easy to solve for the number of plants that have green leaves but not large ones. This is merely . We find this by eliminating the large-leaved plants from the green ones (by subtracting the overlap from the green ones).

### Example Question #1 : Data Analysis And Probability

In a group of  people,  have books,  have pens, and  have neither books nor pens. How many people in the group have only books?

Cannot be determined

Explanation:

Based on the information given, you can draw the following Venn Diagram:

Now, you must begin by solving for . You know that the two circles together will have  in them. This is arrived at by subtracting the people who have neither books nor pens () from the "universe" of people in the sample space (). Now, we know that . This is because of the overlap of  in both groups. We have to get rid of one instance of that. Thus we can solve for :

Now, we can find the number of people with only books by subtracting  from the  to get .

### Example Question #1 : Venn Diagrams

Examine the above Venn diagram. Let  be the universal set of the Presidents of the United States.  is the set of all Presidents born in Virginia;  is the set of all Presidents born after 1850;  is the set of all Presidents whose first name was or is James.

James Abram Garfield was born in Ohio in 1831. In which region would he fall?

II

V

III

I

IV

V

Explanation:

Carter would not fall in set A, since he was not a President born in Virginia.

He would not fall in B, since he was born before 1850.

He would fall in C, since his first name is James.

He would fall in the region included in set C, but not A or B - this is Region V.

### Example Question #1 : Venn Diagrams

Examine the above Venn diagram. Let  be the universal set of the Presidents of the United States.  is the set of all Presidents born in Virginia;  is the set of all Presidents born after 1850;  is the set of all Presidents whose first name was or is James.

James Earl Carter was born in Georgia in 1924. In which region would he fall?

V

II

IV

I

III

III

Explanation:

Carter would not fall in set A, since he was not a President born in Virginia.

He would fall in B, since he was born after 1850.

He would fall in C, since his first name is James.

He would fall in the region included in sets B and C, but not A - this is Region III.

### Example Question #1 : Data Analysis And Probability

Examine the above Venn diagram. Let universal set  represent the set of all words in the English language.

Let  be the set of all words whose last letter is a consonant. Let  be the set of all words whose first letter is a vowel. Let  be the set of all words exactly five letters in length.

Which of the following would be a subset of the set represented by the shaded region in the diagram?

Note: for purposes of this question, "Y" is considered a consonant.

{eagle, uvula, apnea, unsee, abide}

{usher, aspen, ester, order, earth}

{price, value, pinna, trove, three}

{catfish, division, rot, status, giving}

{potato, tomato, breeze, mimosa, magnolia}

{price, value, pinna, trove, three}

Explanation:

The subset must comprise words that fall inside set , but neither nor .

Therefore, all of the words in the subset must have exactly five letters, but cannot begin with a vowel or end with a consonant - that is, we are looking for a set of five-letter words that begin with a consonant and end with a vowel.

The only set among the five choices that matches this description is the set

{price, value, pinna, trove, three}.

### Example Question #1 : Venn Diagrams

Examine the above Venn diagram. Let universal set  represent the set of all words in the English language.

Let  be the set of all words whose last letter is a vowel. Let  be the set of all words whose first letter is a consonant. Let  be the set of all words exactly six letters in length.

Which of the following would be a subset of the set represented by the shaded region in the diagram?

Note: for purposes of this question, "Y" is considered a consonant.

{autism, enough, ideals, occult, unduly}

{tomato, potato, ravine, cabana, marine}

{autistic, estrogen, ideology, opal, understand}

{plateau, portmanteau, calliope, marionette, taco}