/*! 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 5 Prove rigorously that if \(A \su... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 5

Prove rigorously that if \(A \subseteq B,\) then \(A \cap B=A\).

Short Answer

Expert verified
The rigorous proof concludes that if \(A \subseteq B,\) then \(A \cap B=A\). The proof was conducted by demonstrating that \(A \subseteq A \cap B\) and \(A \cap B \subseteq A\), which combine to give the required equality \(A=A \cap B\).

Step by step solution

01

Definition of Subset

Recall the definition of a subset: A set \(A\) is a subset of a set \(B\) (denoted as \(A \subseteq B\)) if every element of \(A\) is also an element of \(B\). So, if \(a\) is an arbitrary element of \(A\), then \(a\) is also in \(B\) because \(A \subseteq B\).
02

Definition of Intersection

Now, recall the definition of the intersection of two sets \(A \cap B\). An element is in \(A \cap B\) if it is in both \(A\) and \(B\). Since \(a\) is in both \(A\) and \(B\) (from the subset definition), \(a\) is in \(A \cap B\). Therefore, every element of \(A\) is in \(A \cap B\). This means \(A \subseteq A \cap B\).
03

Final Proof

We only need to prove \(A \cap B \subseteq A\) for the desired equality. By definition, if \(x\) is an element of \(A \cap B\), \(x\) must be in both \(A\) and \(B\). Hence, \(x\) is in \(A\), and so we have \(A \cap B \subseteq A\). Combining this with the previous step, we have shown \(A = A \cap B\).

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.

Understanding Subsets
In set theory, the concept of a subset is foundational. When we say that set \(A\) is a subset of set \(B\) (written as \(A \subseteq B\)), we mean that every element in \(A\) is also present in \(B\).
Imagine \(B\) as a large basket that can hold everything in \(A\) and possibly more.
If this is true, there is no element in \(A\) that isn't found in \(B\).
  • Example: If \(A = \{1, 2\}\) and \(B = \{1, 2, 3\}\), then \(A\) is a subset of \(B\) because both 1 and 2 are in \(B\).
  • Visualize subsets as smaller circles completely within bigger circles (Venn diagrams are helpful for this).
Understanding subsets helps simplify relationships between different sets you work with in mathematics.
Exploring Intersection of Sets
The intersection of two sets, denoted as \(A \cap B\), is all about finding common elements. It's like looking at two groups and asking, "Who is in both groups?"
For example, if set \(A = \{1, 2\}\) and set \(B = \{2, 3\}\), then the intersection \(A \cap B\) equals \(\{2\}\).
This is because 2 is the only number in both \(A\) and \(B\).
  • When calculating intersections, think of overlapping parts in a Venn diagram where the circles intersect.
  • An intersection is always a set itself, and it includes only the elements that exist in both sets being considered.
Intersections help find shared characteristics between different categories or collections.
Elements of a Set
An element in set theory refers to a single object or member of a set. When you say an item belongs to a set \(A\), you generally write it as \(x \in A\).
This symbol, \(\in\), stands for "is an element of."
If you think of a set as a collection, each element would be like a unique item in that collection.
  • For example, if \(A = \{orange, apple, banana\}\), each fruit is an element of \(A\).
  • Understanding elements is crucial when talking about subsets and intersections, as these concepts often revolve around which elements belong to which sets.
Recognizing elements helps clarify the makeup and structure of any set you're dealing with.

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

Show that if \(A\) has \(n\) elements, then \(\wp(A)\) has \(2^{n}\) elements.

Prove that there is at most one empty set, i.e., show that if \(A\) and \(B\) are sets without elements, then \(A=B\).

Prove that if \(A \subsetneq B,\) then \(B \backslash A \neq \varnothing\).

Prove that if \(A \subseteq B\), then \(A \cup B=B\).

List all elements of \(\\{1,2,3\\}^{3}\).
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.