### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #1 : Venn Diagrams

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?

**Possible Answers:**

**Correct answer:**

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 : Data Analysis

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

What is ?

**Possible Answers:**

**Correct answer:**

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 #1 : Data Analysis

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 ?

**Possible Answers:**

**Correct answer:**

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 #2 : Venn Diagrams

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 ?

**Possible Answers:**

**Correct answer:**

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 #2 : How To Use A Venn Diagram

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

**Possible Answers:**

**Correct answer:**

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 #1 : How To Use A Venn Diagram

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?

**Possible Answers:**

**Correct answer:**

First, calculate the number of students with a dog:

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