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Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: • real numbers, • complex numbers, C • integers, • All students in our class Defining Sets Sets can be defined directly: e.g. {1,2,4,8,16,32,…}, {CSC1130,CSC2110,…} Order, number of occurence are not important. e.g. {A,B,C} = {C,B,A} = {A,A,B,C,B} A set can be an element of another set. {1,{2},{3,{4}}} Defining Sets by Predicates The set of elements, x, in A such that P(x) is true. {}x APx| ( ) The set of prime numbers: Commonly Used Sets • N = {0, 1, 2, 3, …}, the set of natural numbers • Z = {…, -2, -1, 0, 1, 2, …}, the set of integers • Z+ = {1, 2, 3, …}, the set of positive integers • Q = {p/q | p Z, q Z, and q ≠ 0}, the set of rational numbers • R, the set of real numbers Special Sets • Empty Set (null set): a set that has no elements, denoted by ф or {}. • Example: The set of all positive integers that are greater than their squares is an empty set. • Singleton set: a set with one element • Compare: ф and {ф} – Ф: an empty set. Think of this as an empty folder – {ф}: a set with one element. The element is an empty set. Think of this as an folder with an empty folder in it. Venn Diagrams • Represent sets graphically • The universal set U, which contains all the objects under consideration, is represented by a rectangle. The set varies depending on which objects are of interest. • Inside the rectangle, circles or other geometrical figures are used to represent sets. • Sometimes points are used to represent the particular elements of the set. U a u V e o i 7 Membership {7, “Albert”, /2, T} xA x is an element of A x is in A Examples: /2 {7, “Albert”,/2, T} /3 {7, “Albert”, /2, T} 14/2 {7, “Albert”,/2, T} 7 2/3 Containment AB A is a subset of B A is contained in B Every element of A is also an element of B. Examples: R, {3}{5,7,3} every set, A A A is a proper subset of B Set Equivalence Two sets are equal if and only if they have the same elements. That is, if A and B are sets, then A and B are equal if and only if x(x A x B). We write A = B if A and B are equal sets. • Example: • Are sets {1, 3, 5} and {3, 5,1} equal? • Are sets {1, 3, 3, 3, 5, 5, 5, 5} and {1, 3, 5} equal? Basic Operations on Sets union: AB ::{|()()} xxA xB Basic Operations on Sets intersection: AB :: { xxAxB | } Basic Operations on Sets difference: A BxxAxB:: { | ( ) ( )} Basic Operations on Sets complement: A:: {xDxA | } D A Cardinality Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by |S|. • Example: – Let A be the set of odd positive integers less than 10. Then |A| = 5. – Let S be the set of letters in the English alphabet. Then |A| = 26. – Null set has no elements, | ф | = 0. 15 Infinite Sets A set is said to be infinite if it is not finite. • Example: The set of positive integers is infinite. 16 Cardinality • Finding the cardinality of |A U B|: |A U B| = |A| + |B| - |A ∩ B | • Example: A = {1,3,5,7,9}, B = {5,7,9,11} |A U B| = |A| + |B| - |A ∩ B | = 5 + 4 – 3 = 6 17 Partitions of Sets Two sets are disjoint if their intersection is empty. A collection of nonempty sets {A1, A2, …, An} is a partition of a set A if and only if A1, A2, …, An are mutually disjoint. Power Sets power set: pwo ()::{|ASSA } pow(ab , ) ab , , a , b , Cartesian Products • Sets are unordered, a different structure is needed to represent an ordered collections – ordered n-tuples. The ordered n-tuple (a1, a2,…, an) is the ordered collection that has a1 as its first element, a2 as its second element, …, and an as its nth element. • Two ordered n-tuples are equal if and only if each corresponding pair of their elements is equal. – (a1, a2,…, an) = (b1, b2,…, bn) if and only if ai = bi for i = 1, 2, …, n 20 Cartesian Products Let A and B be sets. The Cartesian product of A and B, denoted by A × B, is the set of all ordered pairs (a, b), where aA and bB . Hence, A × B = {(a,b)| aA Λ bB }. • Example: What is the Cartesian product of A = {1,2} and B = {a,b,c}? Solution: A × B = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)} • Cartesian product of A × B and B × A are not equal, unless A = ф or B = ф (so that A × B = ф ) or A = B. B × A = {(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)} 21 Cartesian products The Cartesian product of sets A1, A2, …, An, denoted by A1 ×A2 ×… × An is the set of ordered n-tuples (a1, a2, …, an), where ai belongs to Ai for i = 1,2, …, n. In other words, A1 ×A2 ×…×An = {(a1, a2, …, an) | ai Ai for i = 1,2, …, n}. • Example: What is the Cartesian product of A × B × C where A= {0,1}, B = {1,2}, and C = {0,1,2}? Solution: A × B × C= {(0,1,0), (0,1,1), (0,1,2), (0,2,0), (0,2,1), (0,2,2), (1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)} 22 Set Identities Distributive Law: Set Identities De Morgan’s Law: Proving Set Identities Proving Set Identities Computer Representation of Sets • Represent a subset A of U with the bit string of length n, where the ith bit in the string is 1 if ai belongs to A and is 0 if ai does not belong to A. • Example: – Let U = {1,2,3,4,5,6,7,8,9,10}, and the ordering of elements of U has the elements in increasing order; that is ai = i. What bit string represents the subset of all odd integers in U? Solution: 10 1010 1010 What bit string represents the subset of all even integers in U? Solution: 01 010 10101 What bit string represents the subset of all integers not exceeding 5 in U? Solution: 11 1110 0000 What bit string represents the complement of the set {1,3,5,7,9}? Solution: 01 0101 0101 27 Logic and Bit Operations • Computers represent information using bits. • A bit is a symbol with two possible values, 0 and 1. • By convention, 1 represents T (true) and 0 represents F (false). • A variable is called a Boolean variable if its value is either true or false. • Bit operation – replace true by 1 and false by 0 in logical operations. Table for the Bit Operators OR, AND, and XOR. x y x ν y x Λ y xy 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 28 Logic and Bit Operations DEFINITION 7 A bit string is a sequence of zero or more bits. The length of this string is the number of bits in the string. • Example: Find the bitwise OR, bitwise AND, and bitwise XOR of the bit string 01 1011 0110 and 11 0001 1101. Solution: 01 1011 0110 11 0001 1101 ------------------- 11 1011 1111 bitwise OR 01 0001 0100 bitwise AND 10 1010 1011 bitwise XOR 29 Set Operations • The bit string for the union is the bitwise OR of the bit string for the two sets. The bit string for the intersection is the bitwise AND of the bit strings for the two sets. • Example: – The bit strings for the sets {1,2,3,4,5} and {1,3,5,7,9} are 11 1110 0000 and 10 1010 1010, respectively. Use bit strings to find the union and intersection of these sets. Solution: Union: 11 1110 0000 V 10 1010 1010 = 11 1110 1010, {1,2,3,4,5,7,9} Intersection: 11 1110 0000 Λ 10 1010 1010 = 10 1010 0000, {1,3,5} 30 Russell’s Paradox Let WS:: Sets | SS so SW SS There is a male barber who shaves all those men, and only those men, who do not shave themselves. Does the barber shave himself?.
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