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2. Given sets X, and Y, use the indicated methods to prove X – (XNY) = X – . (a). Algebraic proof (Indicate the laws used. (b). Element argument

2. Given sets X, and Y, use the indicated methods to prove X – (XNY) = X – . (a). Algebraic proof (Indicate the laws used. (b). Element argument

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A full step by step solution please to the attached image hwk question. Both part a and b.

**Problem 2**: Given sets \( X \) and \( Y \), use the indicated methods to prove \( X - (X \cap Y) = X - Y \). **(a) Algebraic proof** (Indicate the laws used). **(b) Element argument** --- In this problem, we need to prove that the set difference between set \( X \) and the intersection of sets \( X \) and \( Y \) is equal to the set difference between sets \( X \) and \( Y \). We will use two approaches: 1. **Algebraic Proof**: - Start with the expression \( X - (X \cap Y) \). - Use set laws and properties (such as distribution, De Morgan’s Laws, etc.) to simplify and prove this equals \( X - Y \). - Clearly indicate which laws are applied at each step. 2. **Element Argument**: - Consider an arbitrary element and show it is included in both sides of the equation. - Use logical reasoning to demonstrate that if an element is in \( X - (X \cap Y) \), it is also in \( X - Y \), and vice versa. This exercise aims to reinforce understanding of set operations and proofs using both algebraic manipulation and logical argumentation.
Transcribed Image Text:**Problem 2**: Given sets \( X \) and \( Y \), use the indicated methods to prove \( X - (X \cap Y) = X - Y \). **(a) Algebraic proof** (Indicate the laws used). **(b) Element argument** --- In this problem, we need to prove that the set difference between set \( X \) and the intersection of sets \( X \) and \( Y \) is equal to the set difference between sets \( X \) and \( Y \). We will use two approaches: 1. **Algebraic Proof**: - Start with the expression \( X - (X \cap Y) \). - Use set laws and properties (such as distribution, De Morgan’s Laws, etc.) to simplify and prove this equals \( X - Y \). - Clearly indicate which laws are applied at each step. 2. **Element Argument**: - Consider an arbitrary element and show it is included in both sides of the equation. - Use logical reasoning to demonstrate that if an element is in \( X - (X \cap Y) \), it is also in \( X - Y \), and vice versa. This exercise aims to reinforce understanding of set operations and proofs using both algebraic manipulation and logical argumentation.
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