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Chapter 2: Problem 119

Describe the three methods used to represent a set. Give an example of a set represented by each method.

Short Answer

Expert verified
Three methods of representing a set include the roster method, set-builder method, and Venn Diagrams. For example, the set of all positive integers less than 6 would be, in the roster method: \(\{1, 2, 3, 4, 5\}\), in set-builder method: \(\{x: x>0 \text{ and } x<6, x \text{ is an integer}\}\), in Venn Diagrams, a circle within a box, with all numbers less than 6 in the box and only \(\{2, 4\}\) in the circle.

Step by step solution

01

Roster Method

Roster method involves listing all the members of the set within braces \(\{ \}\). Elements are separated by commas. For instance, let's represent the set of all positive integers less than 6. It is represented as \(\{1, 2, 3, 4, 5\}\).
02

Set-Builder Method

In the set-builder method, all the elements share a common property, which is specified instead of listing elements. The property is written within braces and after a vertical line or a colon. For example, the set of all positive integers less than 6 in set-builder form can be represented as \(\{x: x>0 \text{ and } x<6, x \text{ is an integer}\}\).
03

Venn Diagrams

Venn diagrams use circles enclosed within a rectangular box. The entirety of the rectangle represents the universal set, whereas each circle represents a subset of the universal set and the elements of the set are written inside. For instance, if the universal set is the set of all numbers less than 6, and the subset is the set of even numbers less than 6, then the Venn Diagram will contain numbers \(\{1, 2, 3, 4, 5\}\) in the box and \(\{2, 4 \}\) in the circle.

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Most popular questions from this chapter

In Exercises 29-32, use the Venn diagram and the given conditions to determine the number of elements in each region, or explain why the conditions are impossible to meet. \(n(U)=38, n(A)=26, n(B)=21, n(C)=18\) \(n(A \cap B)=17, n(A \cap C)=11, n(B \cap C)=8\) \(n(A \cap B \cap C)=7\)

Let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \((B \cup C)^{\prime} \cap A\)

When filling in cardinalities for regions in a two-set Venn diagram, the innermost region, the intersection of the two sets, should be the last region to be filled in.

Let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} \end{aligned} $$ Find each of the following sets. \((A \cap B) \cup(A \cap C)\)

Let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} \end{aligned} $$ Find each of the following sets. \(\left(C^{\prime} \cap A\right) \cup\left(C^{\prime} \cap B^{\prime}\right)\)
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