12.01 Subsets and sample spaces
Introduction
In 7th grade, we were introduced to sample spaces and represented them in lists, tables, and tree diagrams. In Algebra 1 lesson 8.01 Two way frequency tables , we represented bivariate data using two-way frequency tables. This lesson will review those concepts and teach you how to represent bivariate data in a new way, using Venn diagrams.
Subsets and sample spaces
There are two important types of visual models we can use when dealing with bivariate data.
Each circle in a Venn diagram represents a particular set or category.
A two-way frequency table also displays how elements are distributed across two sets:
| A | \text{Not }A | \text{Total} | |
|---|---|---|---|
| B | 5 | 10 | 15 |
| \text{Not }B | 12 | 13 | 25 |
| \text{Total} | 17 | 23 | 40 |
A collection of items is called a set. There are many situations which we can describe and explore mathematically using sets, and to do so we will make use of some new terminology.
A set can be described by listing its elements inside a pair of braces, and we call this set notation. For example, if the set A is "the set of positive integers smaller than nine", we can write this asA = \left\{1, 2, 3, 4, 5, 6, 7, 8\right\}
If there are too many elements to write out but there is a clear pattern to the elements, we can use three dots to indicate that a pattern continues. For example, if the set B is "the set of even whole numbers", we can write this as B = \left\{2, 4, 6, 8, 10, \ldots \right\}
A special set, called the empty set, is the set which contains no elements. It is usually represented by the symbol \emptyset, but can also be expressed in set notation as\emptyset = \left\{\right\}
To summarize the images above, we can use Venn diagrams to represent sets. The large outer rectangle represents the sample space, and each circle in the Venn diagram represents a set. We can also see their intersection as the region where the circles overlap, and their union as the collection of all parts of each circle.
We can also see these components of sets in two-way tables:
| A | \text{Not }A | \text{Total} | |
|---|---|---|---|
| B | w | x | w + x |
| \text{Not }B | y | z | y + z |
| \text{Total} | w + y | x + z | w + x + y + z |
In the table above, the number of elements in set A would be the total w + y, while the number of elements in the complement of A would be x + z. We get similar results for B and its complement by looking at the rows.
The number of elements in the intersection A \cap B would be w. The number of elements in the union A \cup B would be w + x + y. There are w + x + y + z elements in total in the sample space.
Examples
Example 1
P is the set of odd numbers between 2 and 16, and Q is the set given by Q = \left\{1, 2, 3, 5, 8, 13, 21\right\}.
Determine the set given by P \cap Q.
Example 2
In the sample spaceS = \left\{\text{square, triangle, rhombus, parallelogram, hexagon, circle, trapezoid, rectangle}\right\}the subset A is "quadrilaterals" and the subset B is "words beginning with the letter t".
Describe the set "not A" using words, then express it using set notation.
Describe the set "B or not A" using words, then express it using set notation.
Example 3
Consider the sets shown in the following Venn diagram:
State the sample space.
Indicate the region which represents A \cap B' on the diagram, and state the elements of this set.
Example 4
Using the following two-way table, determine the number of elements in the set given by X \cup Y\rq.
| Y | Y\rq | \text{Total} | |
|---|---|---|---|
| X | 82 | 12 | 94 |
| X\rq | 41 | 53 | 94 |
| \text{Total} | 123 | 65 | 188 |
Example 5
One hundred students in a school are asked about the subjects that they study. 58 of them are studying both math and science, 12 are studying math but not science, and 23 are not studying either math or science.
Create a model to represent the sample space. Explain why you chose that model.
Use your model from part (a) to determine how many students are in the set of students studying science.
When working with bivariate data, we often want to find new subsets of a sample space, S, by combining portions of two existing subsets (A and B) using intersections (\cap), unions (\cup), complements (A\rq or B\rq), or some combination of these. Venn Diagrams and two-way frequency tables can help us visualize the various subsets to answer questions.
