If A and B are sets, several other sets can be constructed from them: the intersection of A and B, АПВ3 (x: хEA and x € B}; /| B N A = A N B. the union of A and B, AUB 3 (x: x Е A or x € B}; // BUA = A U B. and the set difference, A but not B, A\B {x: x € A and x ¢ B}. // Is B \ A = A \ B? // The set A \ B is sometimes called the "relative complement" of B in A. When A N B = Ø, sets A and B are said to be disjoint. /| A and B have no common element. The number of elements in a set S is called the cardinality of S and denoted by |S|. When this is a finite number, then |S| e N, and when |S = n, we’ll say that S is an n-set. For any pair of sets, |AUB| = |A|+|B| – |AN B|, and when A and B are disjoint, |AUB| = |A|+ |B|. // since AN B = Ø Furthermore, we always have |AUB| = |A \ B| +|B \ A| + |A N B|. Example 2.1.1: Operations, Sizes, and Subsets Suppose A is the set of odd integers less than 10 and B is the set of primes less than 10. Then = {2,3,5,7} A = {1,3,5,7,9} and ANB = {3,5,7} and AUB = {1,2,3,5,7,9} A \ B = {1,9} and B \ A = {2}; - |AN B| = 5+4 – 3 = |A \ B| + |B \ A| + |A N B| = 2+1+3. 6 = |AUB| = |A| + |B|
If A and B are sets, several other sets can be constructed from them: the intersection of A and B, АПВ3 (x: хEA and x € B}; /| B N A = A N B. the union of A and B, AUB 3 (x: x Е A or x € B}; // BUA = A U B. and the set difference, A but not B, A\B {x: x € A and x ¢ B}. // Is B \ A = A \ B? // The set A \ B is sometimes called the "relative complement" of B in A. When A N B = Ø, sets A and B are said to be disjoint. /| A and B have no common element. The number of elements in a set S is called the cardinality of S and denoted by |S|. When this is a finite number, then |S| e N, and when |S = n, we’ll say that S is an n-set. For any pair of sets, |AUB| = |A|+|B| – |AN B|, and when A and B are disjoint, |AUB| = |A|+ |B|. // since AN B = Ø Furthermore, we always have |AUB| = |A \ B| +|B \ A| + |A N B|. Example 2.1.1: Operations, Sizes, and Subsets Suppose A is the set of odd integers less than 10 and B is the set of primes less than 10. Then = {2,3,5,7} A = {1,3,5,7,9} and ANB = {3,5,7} and AUB = {1,2,3,5,7,9} A \ B = {1,9} and B \ A = {2}; - |AN B| = 5+4 – 3 = |A \ B| + |B \ A| + |A N B| = 2+1+3. 6 = |AUB| = |A| + |B|
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
Problem 1PE
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Linked List
When a set of items is organized sequentially, it is termed as list. Linked list is a list whose order is given by links from one item to the next. It contains a link to the structure containing the next item so we can say that it is a completely different way to represent a list. In linked list, each structure of the list is known as node and it consists of two fields (one for containing the item and other one is for containing the next item address).
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Let A be the set {1,3,5,7,9} and B be the set {1,2,4,8} . Find |A⋃B|
![**2.1.2 Operations on Sets and Cardinality**
If \( A \) and \( B \) are sets, several other sets can be constructed from them:
- The **intersection** of \( A \) and \( B \), \( A \cap B = \{ x : x \in A \text{ and } x \in B \} \);
- \( B \cap A = A \cap B \).
- The **union** of \( A \) and \( B \), \( A \cup B = \{ x : x \in A \text{ or } x \in B \} \);
- \( B \cup A = A \cup B \).
- The **set difference**, \( A \) but not \( B \), \( A \setminus B = \{ x : x \in A \text{ and } x \notin B \} \).
- Is \( B \setminus A = A \setminus B \)?
// The set \( A \setminus B \) is sometimes called the "relative complement" of \( B \) in \( A \).
When \( A \cap B = \emptyset \), sets \( A \) and \( B \) are said to be **disjoint**.
// \( A \) and \( B \) have no common element.
The number of elements in a set \( S \) is called the **cardinality** of \( S \) and denoted by \( |S| \). When this is a finite number, then \( |S| \in \mathbb{N} \), and when \( |S| = n \), we’ll say that \( S \) is an **n-set**. For any pair of sets,
\[
|A \cup B| = |A| + |B| - |A \cap B|,
\]
and when \( A \) and \( B \) are disjoint,
\[
|A \cup B| = |A| + |B|. \quad // \text{since } A \cap B = \emptyset
\]
Furthermore, we always have
\[
|A \cup B| = |A \setminus B| + |B \setminus A| + |A \cap B|.
\]
---
**Example 2.1.1: Operations, Sizes, and Sub](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4792360-22b4-40c0-a48a-02ce03c9f6f7%2F96533def-b713-4e41-90dc-d70b397986cd%2F95zx7cc_processed.png&w=3840&q=75)
Transcribed Image Text:**2.1.2 Operations on Sets and Cardinality**
If \( A \) and \( B \) are sets, several other sets can be constructed from them:
- The **intersection** of \( A \) and \( B \), \( A \cap B = \{ x : x \in A \text{ and } x \in B \} \);
- \( B \cap A = A \cap B \).
- The **union** of \( A \) and \( B \), \( A \cup B = \{ x : x \in A \text{ or } x \in B \} \);
- \( B \cup A = A \cup B \).
- The **set difference**, \( A \) but not \( B \), \( A \setminus B = \{ x : x \in A \text{ and } x \notin B \} \).
- Is \( B \setminus A = A \setminus B \)?
// The set \( A \setminus B \) is sometimes called the "relative complement" of \( B \) in \( A \).
When \( A \cap B = \emptyset \), sets \( A \) and \( B \) are said to be **disjoint**.
// \( A \) and \( B \) have no common element.
The number of elements in a set \( S \) is called the **cardinality** of \( S \) and denoted by \( |S| \). When this is a finite number, then \( |S| \in \mathbb{N} \), and when \( |S| = n \), we’ll say that \( S \) is an **n-set**. For any pair of sets,
\[
|A \cup B| = |A| + |B| - |A \cap B|,
\]
and when \( A \) and \( B \) are disjoint,
\[
|A \cup B| = |A| + |B|. \quad // \text{since } A \cap B = \emptyset
\]
Furthermore, we always have
\[
|A \cup B| = |A \setminus B| + |B \setminus A| + |A \cap B|.
\]
---
**Example 2.1.1: Operations, Sizes, and Sub
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