/*! 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 1 Prove the following relations. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 1

Prove the following relations. $$ E F \subset E \subset E \cup F $$

Short Answer

Expert verified
In conclusion, we have proven the desired relations by showing that every element in the intersection of sets E and F is also an element of set E (\(E \cap F \subseteq E\)), and that every element in set E is also an element of the union of sets E and F (\(E \subseteq E \cup F\)). Therefore, \[E \cap F \subseteq E \subseteq E \cup F\].

Step by step solution

01

Define the set intersection, subset, and set union

Before we prove the desired relations, let's first define the set intersection, subset, and set union. - The intersection of two sets E and F, denoted as \(E \cap F\), is the set of all elements that are present in both sets E and F. - A set A is considered a subset of another set B, denoted as \(A \subseteq B\), if every element of set A is also an element of set B. - The union of two sets E and F, denoted as \(E \cup F\), is the set of all elements that are present in either set E or set F, or both. Now that we've defined these fundamental set operations, we can proceed to prove the relations.
02

Prove that \(E \cap F \subseteq E\)

Let's first prove that the intersection of sets E and F is a subset of E. That is, \(E \cap F \subseteq E\). By definition, an element x is in the intersection of E and F if and only if x is in both sets E and F (\(x \in E \cap F \Leftrightarrow x \in E \,\, \& \,\, x \in F\)). Thus, if an element x is in \(E \cap F\), it must be in set E. This implies that every element in the intersection of E and F is also an element of E, so \(E \cap F \subseteq E\).
03

Prove that \(E \subseteq E \cup F\)

Now, we need to prove that set E is a subset of the union of sets E and F. That is, \(E \subseteq E \cup F\). By the definition of set union, an element x is in the union of E and F if and only if x is in set E, or x is in set F, or both (\(x \in E \cup F \Leftrightarrow x \in E \,\, \text{or} \,\, x \in F \,\, \text{or} \,\, x \in E \cap F\)). Thus, if an element x is in set E, it must also be a member of the union of sets E and F. This implies that every element in set E is also an element of the union of sets E and F, so \(E \subseteq E \cup F\). In conclusion, we have proven the desired relations: \[E \cap F \subseteq E \subseteq E \cup F\]

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.

Set Intersection
In set theory, the concept of "set intersection" is quite important. It involves identifying elements common to both sets being considered.
When we say "intersection," we are essentially looking for shared components.
Mathematically, if we have two sets, say \( E \) and \( F \), their intersection is denoted as \( E \cap F \). This means any element, let's call it \( x \), that belongs to both \( E \) and \( F \) will also belong to the intersection.
  • For example, if \( E = \{1, 2, 3\} \) and \( F = \{2, 3, 4\} \), then \( E \cap F = \{2, 3\} \).
  • It's symbolically expressed as \( \{ x \mid x \in E \text{ and } x \in F \} \).

The intersection operation is very useful when we want to determine what two groups have in common. It鈥檚 a fundamental operation in mathematics that helps in solving various problems concerning collections of objects.
Set Union
The "set union" is another crucial operation in set theory. It involves combining elements from multiple sets to form a new set.
In simple terms, the union of sets \( E \) and \( F \), denoted as \( E \cup F \), contains all elements that are in \( E \), in \( F \), or in both.
  • This means it includes every unique element from both sets.
  • For instance, let \( E = \{1, 2, 3\} \) and \( F = \{3, 4, 5\} \). Their union, \( E \cup F \), would be \( \{1, 2, 3, 4, 5\} \).

Union is especially useful when we aim to capture the entirety of elements under consideration from different particles or groups, without repetition of shared elements. It is a basic but powerful concept that is utilized frequently in areas requiring combination or totality of elements.
Subset
A "subset" is a fundamental idea indicating the relationship between two sets. If we refer to a set \( A \) as a subset of another set \( B \), it means every element in \( A \) is also contained in \( B \), which is symbolically written as \( A \subseteq B \).
  • This can also be true if \( A \) is exactly equal to \( B \), meaning they contain the same elements.
  • For instance, if \( A = \{1, 2\} \) and \( B = \{1, 2, 3\} \), \( A \) is a subset of \( B \) because 1 and 2 are elements of \( B \).

Understanding subsets is essential in many mathematical disciplines, such as combinatorics and computer science, where it helps to analyze relationships and manage groupings or data structures effectively.
It's also important to note the concept of a "proper subset," which implies that \( A \) is a subset of \( B \) but not equal to \( B \) (written as \( A \subset B \)), adding more context when comparing two 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

A certain town of population size 100,000 has 3 newspapers: I, II, and III. The proportions of townspeople that read these papers are as follows: I: 10 percent \(\quad\) I and II: 8 percent \(\quad\) I and II and III: 1 percent \(\begin{aligned} \text { II: } 30 \text { percent } & \text { I and III: } 2 \text { percent } \\ \text { III: } 5 \text { percent } & \text { II and III: } 4 \text { percent } \end{aligned}\) (The list tells us, for instance, that 8000 people read newspapers \(I\) and II.) (a) Find the number of people reading only one newspaper. (b) How many people read at least two newspapers? (c) If \(\mathrm{I}\) and III are moming papers and II is an evening paper, how many people read at least one morning paper plus an evening paper? (d) How many people do not read any newspapers? (e) How many people read only one moming paper and one evening paper?

A die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample space of this experiment? Let \(E_{n}\) denote the event that \(n\) rolls are necessary to complete the experiment. What points of the sample space are contained in \(E_{n}\) ? What is \(\left(\bigcup_{1}^{x} E_{n}\right)^{c} ?\)

Prove the following relations. $$ \left(\bigcup_{1} E_{i}\right) F=\bigcup_{1}^{\infty} E_{i} F, \text { and }\left(\bigcap_{1}^{\infty} E_{i}\right) \cup F=\bigcap_{1}^{\infty}\left(E_{l} \cup F\right) $$

Poker dice is played by simultaneously rolling 5 dice. Show that (a) \(P\\{\) no two alike \(\\}=.0926\) (b) \(P\) (one pair \(\\}=.4630\) (c) \(P\\{\) two pair \(\\}=.2315\) (d) \(P\\{\) three alike \(\\}=.1543\); (e) \(P\\{\) full house \(\\}=.0386\); (f) \(P\) (four alike \(\\}=0193\) (g) \(P\\{\) five alike \(\\}=.0008\).

A hospital administrator codes incoming patients suffering gunshot wounds according to whether they have insurance (coding 1 if they do and 0 if they do not) and according to their condition, which is rated as good (g), fair (f), or serious (s). Consider an experiment that consists of the coding of such a patient. (a) Give the sample space of this experiment. (b) Let \(A\) be the event that the patient is in serious condition. Specify the outcomes in \(A\). (c) Let \(B\) be the event that the patient is uninsured. Specify the outcomes in \(B\). (d) Give all the outcomes in the event \(B^{c} \cup A\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.