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Chapter 2: Problem 77

Find the cardinal number for each set. \(B=\\{x \mid x \in \mathbf{N}\) and \(2 \leq x<7\\}\)

Short Answer

Expert verified
The cardinal number of the set \(B\) is 5.

Step by step solution

01

Identify the set range

According to the provided condition, the set \(B\) consists of natural numbers equal to or greater than 2, and less than 7.
02

Enumerate the members of the set

The natural numbers that fulfill the condition are {2, 3, 4, 5, 6}. It is important to note that the upper limit 7 is not included in the set.
03

Determine the cardinal number of the set

The cardinal number of a set is the number of elements contained in the set. Here, there are five elements in the set \(B\), so the cardinal number of the set \(B\) is 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Numbers
Natural numbers are essentially the numbers we use for counting, starting from 1 and going onwards indefinitely. They form the backbone of arithmetic and are a fundamental concept in mathematics. Here are a few key points about natural numbers:
  • They start at 1 and have no upper bound, extending infinitely in a positive direction.
  • These are the simplest forms of whole numbers, making them easier to work with, especially in basic arithmetic operations such as addition and subtraction.
They include numbers like 1, 2, 3, 4, etc. In some definitions, the number 0 is also included, but in this exercise, we start from 1.
When working with set theory, natural numbers become very important, as they are often used to define ranges, such as in the set given in the exercise, where we look at numbers between 2 and 7.
Set Theory
Set theory is a branch of mathematical logic that studies sets, which are collections of objects. It provides a fundamental language and framework for modern mathematics. Understanding set theory is crucial because it's used to build almost all other areas of mathematics, and each set can be described by its elements.
  • A set is typically defined by listing its elements or by stating a property that its members must satisfy, like in the exercise where set \(B\) consists of all natural numbers between 2 and 7.
  • Sets are usually denoted using curly braces, for example, \(\{2, 3, 4, 5, 6\}\).
In our exercise, identifying and understanding the boundaries of the set \(B\) is critical. We should remember that each number is a unique element in the set.
The study of set theory helps in understanding how to ascertain the number of elements in a set, or its cardinal number, which is integral when evaluating or comparing sets.
Enumeration of Sets
Enumeration of a set involves listing all the elements that the set contains. It is a straightforward process when dealing with finite sets, which have a countable number of elements. Let's take a closer look:
  • When enumerating, it's vital to pay attention to the conditions defining the set, such as in the exercise where numbers have to be between 2 and 7.
  • Enumerated sets are practical because they make the elements visual and manageable, which helps in further calculating properties like the cardinal number.
In the exercise, the enumeration led us to identify \(\{2, 3, 4, 5, 6\}\) as members of set \(B\).
Once a set is enumerated, it becomes easier to identify properties of the set, compute the cardinal number, and perform operations such as union, intersection, or difference with other sets.

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Most popular questions from this chapter

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. \((A \cup B) \cap(A \cup C)\)

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. \((A \cup B)^{\prime} \cap C\)

In Exercises 41-66, 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. \(A \cap B\)

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. \((A \cap B) \cup(A \cap C)\)

In Exercises 164-167, assume \(A \neq B\). Draw a Venn diagram that correctly illustrates the relationship between the sets. \(A \cap B=A\)
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