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**Instructor: Hayk Melikya melikyan@nccu.edu**

Sets Section 6.1 Basic Definitions of Sets Theory Section 6.2 Properties of sets Section 6.3 Proofs and Boolean Algebras The most fundamental notion in all of mathematics is that of a set. We say that a set is a specified collection of objects, called elements (or members) of the set. We denote sets by capital letters A, B, … and elements by lower case letters, like x, y , … and so on. If an element x belongs to a set A, we denote this by x A, if not we write x A. The slides for this text are organized into chapters. This lecture covers Chapter 1. Chapter 1: Introduction to Database Systems Chapter 2: The Entity-Relationship Model Chapter 3: The Relational Model Chapter 4 (Part A): Relational Algebra Chapter 4 (Part B): Relational Calculus Chapter 5: SQL: Queries, Programming, Triggers Chapter 6: Query-by-Example (QBE) Chapter 7: Storing Data: Disks and Files Chapter 8: File Organizations and Indexing Chapter 9: Tree-Structured Indexing Chapter 10: Hash-Based Indexing Chapter 11: External Sorting Chapter 12 (Part A): Evaluation of Relational Operators Chapter 12 (Part B): Evaluation of Relational Operators: Other Techniques Chapter 13: Introduction to Query Optimization Chapter 14: A Typical Relational Optimizer Chapter 15: Schema Refinement and Normal Forms Chapter 16 (Part A): Physical Database Design Chapter 16 (Part B): Database Tuning Chapter 17: Security Chapter 18: Transaction Management Overview Chapter 19: Concurrency Control Chapter 20: Crash Recovery Chapter 21: Parallel and Distributed Databases Chapter 22: Internet Databases Chapter 23: Decision Support Chapter 24: Data Mining Chapter 25: Object-Database Systems Chapter 26: Spatial Data Management Chapter 27: Deductive Databases Chapter 28: Additional Topics Instructor: Hayk Melikya

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Specifying set There are various ways to specify a set. For the set of natural numbers less than or equal to 5, you could write {1, 2, 3, 4, 5} . For sets that cannot be specified by a list, we describe the elements by some property common to the elements in the set but no others, such as in the description A = { x | P (x)} which reads “the set of all x such that the condition P(x) is true.”

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Common Sets:

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Subsets We say that a set A is a subset of a set B if every element of A is also an element of B. Symbolically, we write this as A B and is read “A is contained in B.” Finally, the notation A B means that A is not a subset of B. Sets are often illustrated by Venn diagrams, where sets are represented as circles and elements of the set are points inside the circle.

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Equality of Sets: Two sets are equal (A = B) if they consist of exactly the same elements. In other words, they are equal if (A = B) if and only if (x) (( xA x B) or (A = B) iff (x) (( xA x B) (xB x A )) another way: (A = B) if and only if (A B B A). Empty Set: The set with no elements is called the empty set (or null set) and denoted by the Greek letter (or sometimes the empty bracket { } )

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**Theorem 1 (Guaranteed Subset) For any set A, we have A.**

Proof : Since the goal is to show x x A our job is done before we begin. The reason being that the hypothesis x of the implication is false, being that contains no elements, hence the proposition is true regardless of the set A. In other words is a subset of any set. END

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**Theorem 2 (Transitive Subsets) Let A, B and C be sets**

Theorem 2 (Transitive Subsets) Let A, B and C be sets If A B and B C then A C . Proof: We will prove the conclusion A C and use the hypothesis as needed. Letting x A the goal is to show x C . Since x A and using the assumption A B , we know x B . But the second hypothesis says B C , and so we know x C . Hence, we have proved A C , which proves the theorem. END

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**Subset and Membership:**

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Power Set P(A) An important set in mathematics is the power set. For every set A, we denote by P(A) the set of all subsets of A. Theorem 3 (Power Set) Let A and B be sets. Then A B if and only if P (A) P(B). Proof: (A B) (P(A) P(B)): We start by letting X P(A) and show X P(B) (and use A B as our “helper”). Letting X P(A) we have X A and hence X B. But this means X P(B) and so we have shown P(A) P(B) . (P(A) P(B)) (A B) : We let x A and show x B . If x A, then {x } P(A) , and since P(A) P(B) we know {x } P(B) . But this means x B and so A B .

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**Sec 6.2 Operations on Sets Union, Intersection and Complement**

In traditional arithmetic and algebra, we carry out the binary operations of + and × on numbers. In logic, we have the analogous binary operations of and on sentences. In set theory we have the binary operations of union and intersection of sets, which in a sense are analogous to the ones in arithmetic and sentential logic. Definition ( Union): The union of two sets A and B, denoted A B , is the set of elements that belong to A or B or both. Symbolically A B = {x | x A x B }

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**Definition ( Intersection):**

The intersection of two sets A and B, denoted A B , is the set of elements that belong to A and B. Symbolically A B = {x | x A x B }

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**Definition( Complement):**

The compliment of A, denoted Ac is the set of elements belonging to the universal set U but not A. Symbolically Ac = {x | x ÎU x A } .

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**Definition (Relative Complement or Difference):**

The relative complement of A in B, denoted, B \ A, is the set of elements in B but not in A. Symbolically B \ A = {x | x B x A } The concepts of union, intersection and relative complement of sets can be illustrated graphically by use of Venn diagrams. Each Venn diagram begins with an oval representing the universal set, a set that contains all elements of in discussion. Then, each set in the discussion is represented by a circle, where elements belonging to more than one set are placed in sections where circles overlap.

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**Venn diagrams for two overlapping sets.**

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**Set Identities Commutative Laws: A B = A B and A B = B A**

Associative Laws: (A B) C = A (B C) and (A B) C = A (B C) Distributive Laws: A (B C) = (A B) (A C) and A (B C) = (A B) (A C) Intersection and Union with universal set: A U = A and A U = U Double Complement Law: (Ac)c = A Idempotent Laws: A A = A and A A = A De Morgan’s Laws: (A B)c = Ac Bc and (A B)c = Ac Bc Absorption Laws: A (A B) = A and A (A B) = A Alternate Representation for Difference: A – B = A Bc Intersection and Union with a subset: if A B, then A B = A and A B = B

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**Exercises Is is true that (A – B) (B – C) = A – C?**

Show that (A B) – C = (A – C) (B – C) Is it true that A – (B – C) = (A – B) – C? Is it true that (A – B) (A B) = A?

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Set Partitioning Two sets are called disjoint if they have no elements in common Theorem: A – B and B are disjoint A collection of sets A1, A2, …, An is called mutually disjoint when any pair of sets from this collection is disjoint A collection of non-empty sets {A1, A2, …, An} is called a partition of a set A when the union of these sets is A and this collection consists of mutually disjoint sets

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**Power Set Power set of A is the set of all subsets of A**

Theorem: if A B, then P(A) P(B) Theorem: If set X has n elements, then P(X) has 2n elements. Proof. (By Induction)

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Boolean Algebra Boolean Algebra is a set of elements together with two operations denoted as + and * and satisfying the following properties: a + b = b + a, a * b = b * a (a + b) + c = a + (b + c), (a * b) *c = a * (b * c) a + (b * c) = (a + b) * (a + c), a * (b + c) = (a * b) + (a * c) a + 0 = a, a * 1 = a for some distinct unique 0 and 1 a + ã = 1, a * ã = 0

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**Exercises Simplify: A ((B Ac) Bc)**

Symmetric Difference: A B = (A – B) (B – A) Show that symmetric difference is associative Are A – B and B – C necessarily disjoint? Are A – B and C – B necessarily disjoint?

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**Russell’s Paradox Set of all integers, set of all abstract ideas**

Consider S = {A, A is a set and A A} Is S an element of S? Barber puzzle: a male barber shaves all those men who do not shave themselves. Does the barber shave himself? Consider S = {A U, A A}. Is S S?

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SET. A set is a collection of elements. Sets are usually denoted by capital letters A, B, Ω, etc. Elements are usually denoted by lower case.

SET. A set is a collection of elements. Sets are usually denoted by capital letters A, B, Ω, etc. Elements are usually denoted by lower case.

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