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Chapter 7: Problem 38

Determine whether the pairs of sets are disjoint. a. \(\varnothing,\\{1,3,5\\}\) b. \(\\{0,1,3,4\\},\\{0,2,5,7\\}\)

Short Answer

Expert verified
The first pair of sets, \(\varnothing\) and \(\{1,3,5\}\), is disjoint as the empty set has no elements. The second pair of sets, \(\{0,1,3,4\}\) and \(\{0,2,5,7\}\), is not disjoint since they have the common element 0.

Step by step solution

01

Identify the Pairs of Sets

There are two pairs of sets given: a. \(\varnothing,\\{1,3,5\\}\) b. \(\\{0,1,3,4\\},\\{0,2,5,7\\}\)
02

Check for Common Elements in the First Pair of Sets

In the first pair of sets, one set is the empty set (\(\varnothing\)) which contains no elements, and the other set is \(\{1,3,5\}\). Since the empty set has no elements, it cannot have any common elements with any other set. Therefore, the first pair of sets is disjoint.
03

Check for Common Elements in the Second Pair of Sets

In the second pair of sets, we have: Set A = \(\{0,1,3,4\}\) and Set B = \(\{0,2,5,7\}\) To determine if these sets are disjoint, we need to check if they have any common elements. We can see that both Set A and Set B have the element 0. Since they have a common element, the second pair of sets is not disjoint. In conclusion, the first pair of sets is disjoint, while the second pair of sets is not disjoint.

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Key Concepts

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

Empty Set
The concept of the empty set, often denoted by the symbol \( \varnothing \), is a foundational idea in set theory. Simply put, the empty set is a set that contains no elements at all. It's like an empty basket, where there's nothing inside to count or to look at. This concept might seem simple, but it plays a crucial role in mathematics.
  • In comparison with other sets, the empty set is unique because no matter what element you consider, it is never part of the empty set.
  • The empty set is a subset of every set, including itself. This may sound strange, but it adheres to the rule that says a set A is a subset of a set B if every element of A is also an element of B.

When considering whether pairs of sets are disjoint, as in the exercise, if one of the sets is an empty set, then this pair is automatically disjoint. That's because there are no elements in an empty set to share with the other set.
Common Elements
In order to decide if two sets are disjoint, we need to determine whether they share any common elements. Common elements are simply those elements that appear in both sets.
  • If any single element can be found in both sets, then the sets are not disjoint since they have something in common.
  • If there are no common elements, then the two sets are disjoint.

For example, in the second pair of sets from the exercise, Set A \(\{0,1,3,4\}\) shares the element "0" with Set B \(\{0,2,5,7\}\). Because of this common element, these two sets are not disjoint. Checking for shared elements requires systematically comparing each element from one set with all elements from the other set. It can be a straightforward process if the sets are small, but it becomes more involved with larger sets.
Set Theory
Set theory is a branch of mathematical logic that studies sets, which are collections of objects. These objects can be anything: numbers, letters, or even other sets. The simplicity and flexibility of set theory make it an important part of mathematics.
  • In set theory, sets are the building blocks for more complex mathematical concepts, such as relations, functions, and more.
  • Understanding the nature of sets, like the empty set and the notion of common elements, is important for solving problems concerning disjoint sets.

The concept of disjoint sets, for instance, stems from the fundamental idea of comparing elements in different sets to see if there's any overlap. This principle is used to define relationships between sets and measure the extent to which they intersect or remain separate. Studying sets can help students grasp abstract concepts and see the logical structures underlying mathematics.

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Most popular questions from this chapter

CoURSE ENROLLMENTS Among 500 freshmen pursuing a business degree at a university, 320 are enrolled in an economics course, 225 are enrolled in a mathematics course, and 140 are enrolled in both an economics and a mathematics course. What is the probability that a freshman selected at random from this group is enrolled in a. An economics and/or a mathematics course? b. Exactly one of these two courses? c. Neither an economics course nor a mathematics course?

In an online survey of 1962 executives from 64 countries conducted by Korn/Ferry International between August and October 2006 , the executives were asked if they would try to influence their children's career choices. Their replies: A (to a very great extent), \(\mathrm{B}\) (to a great extent), \(\mathrm{C}\) (to some extent), D (to a small extent), and \(\mathrm{E}\) (not at all) are recorded below: $$ \begin{array}{lccccc} \hline \text { Answer } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 135 & 404 & 1057 & 211 & 155 \\ \hline \end{array} $$ What is the probability that a randomly selected respondent's answer was \(\mathrm{D}\) (to a small extent) or \(\mathrm{E}\) (not at all)?

Let \(S=\left\\{s_{1}, s_{2}, s_{3}, s_{4}, s_{5}\right\\}\) be the sample space associated with an experiment having the following probability distribution: $$ \begin{array}{lccccc} \hline \text { Outcome } & s_{1} & s_{2} & s_{3} & s_{4} & s_{5} \\ \hline \text { Probability } & \frac{1}{14} & \frac{3}{14} & \frac{6}{14} & \frac{2}{14} & \frac{2}{14} \\ \hline \end{array} $$ Find the probability of the event: a. \(A=\left\\{s_{1}, s_{2}, s_{4}\right\\}\) b. \(B=\left\\{s_{1}, s_{5}\right\\}\) c. \(C=S\)

In an attempt to study the leading causes of airline crashes, the following data were compiled from records of airline crashes from 1959 to 1994 (excluding sabotage and military action). $$ \begin{array}{lc} \hline \text { Primary Factor } & \text { Accidents } \\ \hline \text { Pilot } & 327 \\ \hline \text { Airplane } & 49 \\ \hline \text { Maintenance } & 14 \\ \hline \text { Weather } & 22 \\ \hline \text { Airport/air traffic control } & 19 \\ \hline \text { Miscellaneous/other } & 15 \\ \hline \end{array} $$ Assume that you have just learned of an airline crash and that the data give a generally good indication of the causes of airline crashes. Give an estimate of the probability that the primary cause of the crash was due to pilot error or bad weather.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The numbers 1,2, and 3 are written separately on three pieces of paper. These slips of paper are then placed in a bowl. If you draw two slips from the bowl, one at a time and without replacement, then the sample space for this experiment consists of six elements.
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