ICOM 5016 Introduction to Database Systems Lecture 2
ICOM 5016 – Introduction to Database Systems Lecture 2 Dr. Manuel Rodriguez Department of Electrical and Computer Engineering University of Puerto Rico, Mayagüez
Objectives n Introduce Set Theory n Describe Entity Relationship Model (E-R) Model n Entity Sets n Relationship Sets n Design Issues n Mapping Constraints n Keys n E-R Diagram n Extended E-R Features n Design of an E-R Database Schema n Reduction of an E-R Schema to Tables Database System Concepts 2. 2 ©Silberschatz, Korth and Sudarshan
On Sets and Relations n A set S is a collection of objects, where there are no duplicates H Examples 4 A = {a, b, c} 4 B = {0, 2, 4, 6, 8} 4 C = {Jose, Pedro, Ana, Luis} n The objects that are part of a set S are called the elements of the set. H Notation: 4 0 is an element of set B is written as 0 B. 4 3 is not an element of set B is written as 3 B. Database System Concepts 2. 3 ©Silberschatz, Korth and Sudarshan
Cardinality of Sets n Sets might have H 0 elements – called the empty set . H 1 elements – called a singleton H N elements – a set of N elements (called a finite set) 4 Ex: S = {car, plane, bike} H elements – an infinite number of elements (called infinite set) 4 Integers, Real, 4 Even numbers: E = {0, 2, 4, 6, 8, 10, …} – Dot notation means infinite number of elements n The cardinality of a set is its number of elements H Notation: cardinality of S is denoted by |S| H Could be an integer number or infinity symbol . Database System Concepts 2. 4 ©Silberschatz, Korth and Sudarshan
Cardinality of Sets (cont. ) n Some examples: H A = {a, b, c} 4|A| = 3 H R – set of real numbers 4|R| = H E = {0, 2, 3, 4, 6, 8, 10, …} 4|E| = H the empty set 4| | = 0 Database System Concepts 2. 5 ©Silberschatz, Korth and Sudarshan
Set notations and equality of Sets n Enumeration of elements of set S H A = {a, b c} H E = {0, 2, 4, 6, 8, 10, …} n Enumeration of the properties of the elements in S H E = {x : x is an even integer} H E = {x: x I and x/2=0, where I is the set of integers. } n Two sets are said to be equal if and if only they both have the same elements H A = {a, b, c}, B = {a, b, c}, then A = B H if C = {a, b, c, d}, then A C 4 Because d A Database System Concepts 2. 6 ©Silberschatz, Korth and Sudarshan
Sets and Subsets n Let A and B be two sets. B is said to be a subsets of A if and only if every member x of B is also a member of A H Notation: B A H Examples: 4 A = {1, 2, 3, 4, 5, 6}, B = {1, 2}, then B A 4 D = {a, e, i, o, u}, F = {a, e, i, o, u}, then F D H If B is a subset of A, and B A, then we call B a proper subset 4 Notation: B A 4 A = {1, 2, 3, 4, 5, 6}, B = {1, 2}, then B A H The empty set is a subset of every set, including itself 4 A, for every set A H If B is not a subset of A, then we write B A Database System Concepts 2. 7 ©Silberschatz, Korth and Sudarshan
Set Union n Let A and B be two sets. Then, the union of A and B, denoted by A B is the set of all elements x such that either x A or x B. H A B = {x: x A or x B} n Examples: H A = {10, 20 , 30, 40, 100}, B = {1, 2 , 10, 20} then A B = {1, 2, 10, 20, 30, 40, 100} H C = {Tom, Bob, Pete}, then C = C H For every set A, A A = A Database System Concepts 2. 8 ©Silberschatz, Korth and Sudarshan
Set Intersection n Let A and B be two sets. Then, the intersection of A and B, denoted by A B is the set of all elements x such that x A and x B. H A B = {x: x A and x B} n Examples: H A = {10, 20 , 30, 40, 100}, B = {1, 2 , 10, 20} then A B = {10, 20} H Y = {red, blue, green, black}, X = {black, white}, then Y X = {black} H E = {1, 2, 3}, M={a, b} then, E M = H C = {Tom, Bob, Pete}, then C = n For every set A, A A = A n Sets A and B disjoint if and only if A B = H They have nothing in common Database System Concepts 2. 9 ©Silberschatz, Korth and Sudarshan
Set Difference n Let A and B be two sets. Then, the difference between A and B, denoted by A - B is the set of all elements x such that x A and x B. H A - B = {x: x A and x B} n Examples: H A = {10, 20 , 30, 40, 100}, B = {1, 2 , 10, 20} then A - B = {30, 40, 100} H Y = {red, blue, green, black}, X = {black, white}, then Y - X = {red, blue, green} H E = {1, 2, 3}, M={a, b} then, E - M = E H C = {Tom, Bob, Pete}, then C - = C H For every set A, A - A = Database System Concepts 2. 10 ©Silberschatz, Korth and Sudarshan
Power Set and Partitions n Power Set: Given a set A, then the set of all possible subsets of A is called the power set of A. H Notation: H Example: 4 A = {a, b, 1} then = { , {a}, {b}, {1}, {a, b}, {a, 1}, {b, 1}, {a, b, 1}} 4 Note: empty set is a subset of every set. n Partition: A partition of a nonempty set A is a subset of H H such that Each set element P is not empty For D, F , D F, it holds that D F = The union of all P is equal to A. Example: A = {a, b, c}, then = {{a, b}, {c}}. Also = {{a}, {b}, {c}}. But this is not: M = {{a, b}, {c}} Database System Concepts 2. 11 ©Silberschatz, Korth and Sudarshan
Cartesian Products and Relations n Cartesian product: Given two sets A and B, the Cartesian product between and A and, denoted by A x B, is the set of all ordered pairs (a, b) such a A and b B. H Formally: A x B = {(a, b): a A and b B} H Example: A = {1, 2}, B = {a, b}, then A x B = {(1, a), (1, b), (2, a), (2, b)}. n A binary relation R on two sets A and B is a subset of A x B. H Example: A = {1, 2}, B = {a, b}, then A x B = {(1, a), (1, b), (2, a), (2, b)}, and one possible R A x B = {(1, a), (2, a)} Database System Concepts 2. 12 ©Silberschatz, Korth and Sudarshan
N-ary Relations n Let A 1, A 2, …, An be n sets, not necessarily distinct, then an n-ary relation R on A 1, A 2, …, An is a sub-set of A 1 x A 2 x … x An. H Formally: R A 1 x A 2 x … x An H R = {(a 1, a 2, …, an) : a 1 A 1 and a 2 A 2 and … and an An} H Example: 4 R = set of all real numbers 4 R x R = three-dimensional space 4 P = {(x, y, z): x R and x 0 and y R and y 0 and y R and y 0} = Set of all three-dimensional points that have positive coordinates Database System Concepts 2. 13 ©Silberschatz, Korth and Sudarshan
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