# ISEE Upper Level Math : Venn Diagrams

## Example Questions

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

Refer to the above Venn diagram.

Define universal set , the set of natural numbers.

Define sets  and  as follows:

Which of the following numbers is an element of the set represented by the gray area in the diagram?

Explanation:

The gray area represents the set of all elements that are in  but not in .

is the set of integers that, when divided by 3, yield remainder 2. Therefore, we can eliminate 102 and 105, both multiples of 3, and 103, which, when divided by 3, yields remainder 1.

is the set of integers that, when divided by 4, yield remainder 1. Since we do not want an element from this set, we can eliminate 101, but not 104.

104 is the correct choice.

### 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 intersection of sets  and  - that is, the set of all elements of  that are elements of both  and . We want all of the letters that fall in both circles, which from the diagram can be seen to be  and . Therefore,

### Example Question #2 : 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. Which of the following is equal to  ?

Explanation:

is the complement of  - the set of all elements in  not in

is the intersection of sets  and  - that is, the set of all elements of  that are elements of both  and . Therefore,  is the set of all elements that are not in both  and , which can be seen from the diagram to be all elements except  and . Therefore,

.

### 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. Which of the following is equal to  ?

Explanation:

is the complement of  - the set of all elements in  not in .

is the union of sets  and  - the set of all elements in either  or . Therefore,  is the set of all elements in neither  nor , which can be seen from the diagram to be only one element - . Therefore,

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

The following Venn diagram depicts the number of students who play hockey, football, and baseball. How many students play football and baseball?

Explanation:

The number of students who play football or baseball can by finding the summer of the number of students who play football alone, baseball alone, baseball and football, and all three sports.

### Example Question #2 : Data Analysis And Probability

A class of students was asked whether they have cats, dogs, or both.The results are depicted in the following Venn diagram. How many students do not have a dog?

Explanation:

First, calculate the number of students with a dog:

Next, subtract the number of students with a dog from the total number of students.