/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Let \(U=\\{1,2,3,4,5,6,7, a, b, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 4

Let \(U=\\{1,2,3,4,5,6,7, a, b, c, d, e\\} .\) If \(A=\\{1,2, a, e\\}\) and \(B=\\{1,2,3,4, a, b, c\\}\), find: a. \(n\left(A^{c}\right)\) b. \(n\left(A \cap B^{c}\right)\) c. \(n\left(A \cup B^{c}\right)\) d. \(n\left(A^{c} \cap B^{c}\right)\)

Short Answer

Expert verified
a) \(n(A^c) = 8\) b) \(n(A \cap B^c) = 1\) c) \(n(A \cup B^c) = 8\) d) \(n(A^c \cap B^c) = 4\)

Step by step solution

01

Find the complements of A and B.

Recall that the complement of a set is the set of all elements in the universal set that are not in the set itself. We can find the complements of A and B by subtracting their elements from the universal set U. \(A^c = U - A = \{3, 4, 5, 6, 7, b, c, d\}\) \(B^c = U - B = \{5, 6, 7, d, e\}\)
02

Calculate the cardinalities of the given set operations.

a) To find the cardinality of \(A^c\), count the number of elements in the set: \(n(A^c) = 8\) b) The intersection \(A \cap B^c\) contains the elements that are present in both A and \(B^c\). We list the elements in both sets and find their intersection: \(A = \{1, 2, a, e\}\) \(B^c = \{5, 6, 7, d, e\}\) So, \(A \cap B^c = \{e\}\) and hence \(n(A \cap B^c) = 1\). c) The union \(A \cup B^c\) contains all the elements that are either in A or in \(B^c\) or in both. List the elements in both sets and find their union: \(A = \{1, 2, a, e\}\) \(B^c = \{5, 6, 7, d, e\}\) So, \(A \cup B^c = \{1, 2, a, e, 5, 6, 7, d\}\) and hence \(n(A \cup B^c) = 8\). d) The intersection \(A^c \cap B^c\) contains the elements that are present in both \(A^c\) and \(B^c\). List the elements in both sets and find their intersection: \(A^c = \{3, 4, 5, 6, 7, b, c, d\}\) \(B^c = \{5, 6, 7, d, e\}\) So, \(A^c \cap B^c = \{5, 6, 7, d\}\) and hence \(n(A^c \cap B^c) = 4\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Cardinality of Sets
The concept of cardinality refers to the number of elements contained within a set. Think of it as a way of counting how many distinct items are in a group. For example, if we have a set that contains apples, oranges, and bananas, the cardinality of that set is 3 because there are three different types of fruit.

When we look at the original exercise involving sets A and B within the universal set U, calculating the cardinality of a set is simply a matter of counting the elements. In the solution provided, the cardinality of the complement of set A, denoted as \(n(A^c)\), is found to be 8. The process used here involves identifying all elements of the universal set that are not in set A and then counting them, which is a straightforward application of the definition of cardinality.
Complement of a Set
Imagine you have a whole pizza (your universal set), and you ate a few slices (your set A). The slices that are left untouched (not in A) represent the complement of set A, denoted as \(A^c\).

In set theory, the complement of a set consists of all the elements that are in the universal set but not in the set itself. In the provided exercise, the complement of set A was calculated by removing the elements of A from the universal set U, resulting in a new set \(A^c\) which contains the elements that are not in A. It's like determining what has not been chosen or included from a complete collection of options.
Intersection of Sets
When two friends share common interests, that's like finding the intersection of two sets. The intersection, symbolized by \(\cap\), represents the set of elements that sets A and B have in common. If A is a set of your favorite movies, and B is your friend's favorites, then \(A \cap B\) would list the movies that both of you love to watch.

In the solution, the intersection \(A \cap B^c\) involved finding which elements were shared between set A and the complement of set B, \(B^c\). In this case, since \(e\) is the only element shared, the intersection consisted solely of that single element, and consequently, it had a cardinality of 1.
Union of Sets
Choosing toppings for a pizza from two different lists mirrors the idea of forming the union of two sets. The union of sets, denoted by \(\cup\), combines all the elements from both sets without any repetition. It's like merging two different playlists into one without duplicating any songs.

In the exercise, the union of set A and the complement of set B, \(A \cup B^c\), was found by listing all unique elements from both sets. The result was a set that included every distinct element from both A and the complement of B, demonstrating the inclusive nature of the union operation. With this union, we see the cardinality of the resulting set as 8, which includes every unique item from the individual sets.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The grade distribution for a certain class is shown in the following table. Find the probability distribution associated with these data. $$ \begin{array}{lccccc} \hline \text { Grade } & \text { A } & \text { B } & \text { C } & \text { D } & \text { F } \\ \hline \text { Frequency of } & & & & & \\ \text { Occurrence } & 4 & 10 & 18 & 6 & 2 \\ \hline \end{array} $$

If a ball is selected at random from an urn containing three red balls, two white balls, and five blue balls, what is the probability that it will be a white ball?

In a survey conducted in 2007 of 1402 workers 18 yr and older regarding their opinion on retirement benefits, the following data were obtained: 827 said that it was better to have excellent retirement benefits with a lower-than-expected salary, 477 said that it was better to have a higher-than-expected salary with poor retirement benefits, 42 said "neither," and 56 said "not sure." If a worker in the survey is selected at random, what is the probability that he or she answered that it was better to have a. Excellent retirement benefits with a lower-than-expected salary? b. A higher-than-expected salary with poor retirement benefits?

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)?

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.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.