### Proof of Set Identity Using the Identities Sets Table#### Problem StatementLet \( A \), \( B \), and \( C \) be sets. Using the Identities sets table, show that:\[\overline{(A \cup B)} \cap \overline{(B \cup C)} \cap \overline{(A \cup C)} = \overline{A} \cap \overline{B} \cap \overline{C}\]#### ProofTo prove the given identity using set laws, we will follow several steps employing De Morgan’s laws and properties of set operations.1. **De Morgan’s Laws**: - \(\overline{A \cup B} = \overline{A} \cap \overline{B}\) - \(\overline{A \cap B} = \overline{A} \cup \overline{B}\)2. **Applying De Morgan’s Law** to each term in the left-hand side of the equation:\[\overline{(A \cup B)} = \overline{A} \cap \overline{B}\]\[\overline{(B \cup C)} = \overline{B} \cap \overline{C}\]\[\overline{(A \cup C)} = \overline{A} \cap \overline{C}\]3. **Substituting these results** into the left-hand side of the given identity:\[\overline{(A \cup B)} \cap \overline{(B \cup C)} \cap \overline{(A \cup C)} = (\overline{A} \cap \overline{B}) \cap (\overline{B} \cap \overline{C}) \cap (\overline{A} \cap \overline{C})\]4. **Simplifying the expression**:Using the associative and commutative properties of intersection:\[ (\overline{A} \cap \overline{B}) \cap (\overline{B} \cap \overline{C}) \cap (\overline{A} \cap \overline{C}) = (\overline{A} \cap \overline{A}) \cap (\overline{B}

Let A, B and C be sets. Using the Identities sets table, show that following expression. we know…

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(BU C) n Let A, B, and C be sets. Using the Identities sets table, show that (A U B) n (AUC) = ĀnBnc %3D

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

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Concept explainers

LUP Decomposition

LU decomposition stands for Lower-Upper decomposition. LU decomposition is the factorization of the lower triangular matrix and an upper triangular matrix. It is very useful in linear algebra and numerical analysis. A permutation matrix is usually included in the product. Gaussian elimination in a matrix form is known as LU decomposition, When inverting a matrix or evaluating its determinant, this is a crucial step. Tadeusz Banachiewicz, a Polish mathematician, first proposed the LU decomposition in 1938.

Question

### Proof of Set Identity Using the Identities Sets Table

#### Problem Statement
Let \( A \), \( B \), and \( C \) be sets. Using the Identities sets table, show that:

\[
\overline{(A \cup B)} \cap \overline{(B \cup C)} \cap \overline{(A \cup C)} = \overline{A} \cap \overline{B} \cap \overline{C}
\]

#### Proof
To prove the given identity using set laws, we will follow several steps employing De Morgan’s laws and properties of set operations.

1. **De Morgan’s Laws**:
- \(\overline{A \cup B} = \overline{A} \cap \overline{B}\)
- \(\overline{A \cap B} = \overline{A} \cup \overline{B}\)

2. **Applying De Morgan’s Law** to each term in the left-hand side of the equation:

\[
\overline{(A \cup B)} = \overline{A} \cap \overline{B}
\]
\[
\overline{(B \cup C)} = \overline{B} \cap \overline{C}
\]
\[
\overline{(A \cup C)} = \overline{A} \cap \overline{C}
\]

3. **Substituting these results** into the left-hand side of the given identity:

\[
\overline{(A \cup B)} \cap \overline{(B \cup C)} \cap \overline{(A \cup C)} = (\overline{A} \cap \overline{B}) \cap (\overline{B} \cap \overline{C}) \cap (\overline{A} \cap \overline{C})
\]

4. **Simplifying the expression**:

Using the associative and commutative properties of intersection:
\[
(\overline{A} \cap \overline{B}) \cap (\overline{B} \cap \overline{C}) \cap (\overline{A} \cap \overline{C}) = (\overline{A} \cap \overline{A}) \cap (\overline{B}

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