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1 Introduction to Abstract Mathematics Sets Section 6.1 Basic Definitions of Sets Theory Section 6.2 Properties of sets Section 6.3 Proofs and Boolean Algebras Instructor: Hayk Melikya melikyan@nccu.edu 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.

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2 Introduction to Abstract Mathematics Specifying set v 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}. v 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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3 Introduction to Abstract Mathematics Common Sets:

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4 Introduction to Abstract Mathematics Subsets v We say that a set A is a subset of a set B if every element of A is also an element of B. v Symbolically, we write this as A B and is read “ A is contained in B.” v Finally, the notation A B means that A is not a subset of B. v 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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5 Introduction to Abstract Mathematics Equality of Sets: v 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) (( x A x B) or (A = B) iff ( x) (( x A x B) (x B 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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6 Introduction to Abstract Mathematics 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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7 Introduction to Abstract Mathematics 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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8 Introduction to Abstract Mathematics Subset and Membership:

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9 Introduction to Abstract Mathematics 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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10 Introduction to Abstract Mathematics Sec 6.2 Operations on Sets Union, Intersection and Complement v 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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11 Introduction to Abstract Mathematics 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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12 Introduction to Abstract Mathematics Definition( Complement): The compliment of A, denoted A c is the set of elements belonging to the universal set U but not A. Symbolically A c = {x | x U x A }.

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13 Introduction to Abstract Mathematics 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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14 Introduction to Abstract Mathematics Venn diagrams for two overlapping sets.

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15 Introduction to Abstract Mathematics

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16 Introduction to Abstract Mathematics

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17 Introduction to Abstract Mathematics

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18 Introduction to Abstract Mathematics

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19 Introduction to Abstract Mathematics

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21 Introduction to Abstract Mathematics

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22 Introduction to Abstract Mathematics Set Identities v Commutative Laws: A B = A B and A B = B A v Associative Laws: (A B) C = A (B C) and (A B) C = A (B C) v Distributive Laws: A (B C) = (A B) (A C) and A (B C) = (A B) (A C) v Intersection and Union with universal set: A U = A and A U = U v Double Complement Law: (A c ) c = A v Idempotent Laws: A A = A and A A = A v De Morgan’s Laws: (A B) c = A c B c and (A B) c = A c B c v Absorption Laws: A (A B) = A and A (A B) = A v Alternate Representation for Difference: A – B = A B c v Intersection and Union with a subset: if A B, then A B = A and A B = B

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23 Introduction to Abstract Mathematics Exercises v Is is true that (A – B) (B – C) = A – C? v Show that (A B) – C = (A – C) (B – C) v Is it true that A – (B – C) = (A – B) – C? v Is it true that (A – B) (A B) = A?

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24 Introduction to Abstract Mathematics Set Partitioning v Two sets are called disjoint if they have no elements in common v Theorem: A – B and B are disjoint v A collection of sets A 1, A 2, …, A n is called mutually disjoint when any pair of sets from this collection is disjoint v A collection of non-empty sets {A 1, A 2, …, A n } 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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25 Introduction to Abstract Mathematics Power Set v Power set of A is the set of all subsets of A v Theorem: if A B, then P(A) P(B) v Theorem: If set X has n elements, then P(X) has 2 n elements. Proof. (By Induction)

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26 Introduction to Abstract Mathematics Boolean Algebra v 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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27 Introduction to Abstract Mathematics Exercises v Simplify: A ((B A c ) B c ) v Symmetric Difference: A B = (A – B) (B – A) v Show that symmetric difference is associative v Are A – B and B – C necessarily disjoint? v Are A – B and C – B necessarily disjoint?

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28 Introduction to Abstract Mathematics Russell’s Paradox v Set of all integers, set of all abstract ideas v Consider S = {A, A is a set and A A} v Is S an element of S? v Barber puzzle: a male barber shaves all those men who do not shave themselves. Does the barber shave himself? v Consider S = {A U, A A}. Is S S?

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