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Prove that the empty set is a subset of every set Se

Prove that the empty set is a subset of every set Se

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Title: Understanding Subsets: The Empty Set Content: In this section, we will explore a fundamental concept in set theory: the relationship between the empty set and other sets. Our objective is to prove the statement: "The empty set is a subset of every set S." **Theorem:** Let \( S \) be any set. The empty set, denoted as \( \emptyset \), is a subset of \( S \). **Proof:** By definition, a set \( A \) is a subset of a set \( B \) if every element of \( A \) is also an element of \( B \). 1. Consider the empty set \( \emptyset \). 2. By definition, the empty set contains no elements. 3. Therefore, there are no elements in \( \emptyset \) that could possibly violate the subset condition. 4. Hence, it is vacuously true that all elements of \( \emptyset \) are contained in any set \( S \). Thus, we conclude that \( \emptyset \subseteq S \) for any set \( S \). This concept is essential in understanding the foundational properties of sets and subsets and is widely used in various mathematical fields.
Transcribed Image Text:Title: Understanding Subsets: The Empty Set Content: In this section, we will explore a fundamental concept in set theory: the relationship between the empty set and other sets. Our objective is to prove the statement: "The empty set is a subset of every set S." **Theorem:** Let \( S \) be any set. The empty set, denoted as \( \emptyset \), is a subset of \( S \). **Proof:** By definition, a set \( A \) is a subset of a set \( B \) if every element of \( A \) is also an element of \( B \). 1. Consider the empty set \( \emptyset \). 2. By definition, the empty set contains no elements. 3. Therefore, there are no elements in \( \emptyset \) that could possibly violate the subset condition. 4. Hence, it is vacuously true that all elements of \( \emptyset \) are contained in any set \( S \). Thus, we conclude that \( \emptyset \subseteq S \) for any set \( S \). This concept is essential in understanding the foundational properties of sets and subsets and is widely used in various mathematical fields.
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