Let A, B, and C be sets. Is the following equality true or false? A x (BUC) = (A × B) U (A × C')
Let A, B, and C be sets. Is the following equality true or false? A x (BUC) = (A × B) U (A × 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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![**Mathematical Set Theory: Cartesian Products and Unions**
**Problem Statement:**
Let \( A, B, \) and \( C \) be sets. Is the following equality true or false?
\[
A \times (B \cup C) = (A \times B) \cup (A \times C)
\]
**Explanation:**
To evaluate this, we need to understand the operations involved:
1. **Union (\(\cup\))**:
- The union of sets \( B \) and \( C \), denoted \( B \cup C \), is the set of elements that are in \( B \), in \( C \), or in both.
2. **Cartesian Product (\(\times\))**:
- The Cartesian product of \( A \) and another set, say \( D \), denoted \( A \times D \), is the set of all ordered pairs \((a, d)\) where \( a \in A \) and \( d \in D \).
**Verification:**
The left side of the equality, \( A \times (B \cup C) \), represents the set of ordered pairs where the first element is from \( A \) and the second element is from \( B \cup C \). Thus, each pair takes the form \((a, x)\), where \( a \in A \) and \( x \in B \cup C \).
The right side, \((A \times B) \cup (A \times C)\), represents the union of two Cartesian products: \( A \times B \), which are pairs \((a, b)\) with \( a \in A \) and \( b \in B \), and \( A \times C \), which are pairs \((a, c)\) with \( a \in A \) and \( c \in C \).
**Conclusion:**
Both sides of the equation describe the same set of ordered pairs. Therefore, the equality is true.
The equality demonstrates that taking the Cartesian product of a set \( A \) with the union of two sets \( B \) and \( C \) is equivalent to taking the union of the Cartesian products of \( A \) with \( B \) and \( A \) with \( C \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F406074af-a425-40d0-923c-17adedf14fc2%2F46c49520-a8e6-4658-bf16-67eeaf67d311%2Fj7pw92a_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematical Set Theory: Cartesian Products and Unions**
**Problem Statement:**
Let \( A, B, \) and \( C \) be sets. Is the following equality true or false?
\[
A \times (B \cup C) = (A \times B) \cup (A \times C)
\]
**Explanation:**
To evaluate this, we need to understand the operations involved:
1. **Union (\(\cup\))**:
- The union of sets \( B \) and \( C \), denoted \( B \cup C \), is the set of elements that are in \( B \), in \( C \), or in both.
2. **Cartesian Product (\(\times\))**:
- The Cartesian product of \( A \) and another set, say \( D \), denoted \( A \times D \), is the set of all ordered pairs \((a, d)\) where \( a \in A \) and \( d \in D \).
**Verification:**
The left side of the equality, \( A \times (B \cup C) \), represents the set of ordered pairs where the first element is from \( A \) and the second element is from \( B \cup C \). Thus, each pair takes the form \((a, x)\), where \( a \in A \) and \( x \in B \cup C \).
The right side, \((A \times B) \cup (A \times C)\), represents the union of two Cartesian products: \( A \times B \), which are pairs \((a, b)\) with \( a \in A \) and \( b \in B \), and \( A \times C \), which are pairs \((a, c)\) with \( a \in A \) and \( c \in C \).
**Conclusion:**
Both sides of the equation describe the same set of ordered pairs. Therefore, the equality is true.
The equality demonstrates that taking the Cartesian product of a set \( A \) with the union of two sets \( B \) and \( C \) is equivalent to taking the union of the Cartesian products of \( A \) with \( B \) and \( A \) with \( C \).
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