11. and C. Find a counter example to show that the following statement need not be true for all sets A, B, (AUB=AUC) → (B = C)
11. and C. Find a counter example to show that the following statement need not be true for all sets A, B, (AUB=AUC) → (B = C)
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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![**Problem 11:** Find a counterexample to show that the following statement need not be true for all sets \( A, B, \) and \( C \):
\[
(A \cup B = A \cup C) \to (B = C)
\]
**Explanation:**
This problem requires finding an instance where the statement \( (A \cup B = A \cup C) \to (B = C) \) does not hold. In set theory, the union of two sets \( A \cup B \) includes all elements that are in \( A \), in \( B \), or in both. The statement means if the union of \( A \) and \( B \) is equal to the union of \( A \) and \( C \), then the sets \( B \) and \( C \) themselves must be equal. A counterexample would demonstrate a situation where this implication is false even though the premise holds true.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F868daa41-93eb-4d05-be30-2d6217bf0561%2F679f61c6-dac6-41ee-83cb-1190ee97de6e%2Fczgxypf_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 11:** Find a counterexample to show that the following statement need not be true for all sets \( A, B, \) and \( C \):
\[
(A \cup B = A \cup C) \to (B = C)
\]
**Explanation:**
This problem requires finding an instance where the statement \( (A \cup B = A \cup C) \to (B = C) \) does not hold. In set theory, the union of two sets \( A \cup B \) includes all elements that are in \( A \), in \( B \), or in both. The statement means if the union of \( A \) and \( B \) is equal to the union of \( A \) and \( C \), then the sets \( B \) and \( C \) themselves must be equal. A counterexample would demonstrate a situation where this implication is false even though the premise holds true.
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