Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103 Chapter 2 Sets Slides are adopted from Discrete Mathematics and It's Applications Kenneth H. Rosen; 7th edition, 2012. Chapter 2.1 Sets 1
Definition : A set is an unordered collection of objects. Definition : The objects in a set are called the elements, or members, of the set. Capital letters (A, B, S ) used for sets Italic lower-case letters (a, x, y ) used to denote elements of sets There are several ways to describe a set. 1. Listing all members of a set, when this is possible Some examples: {1, 2, 3} is the set containing 1 and 2 and 3. {1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant. {1, 2, 3} = {3, 2, 1} since sets are unordered. {1, 2, 3, } is a way we denote an infinite set. = { } is the empty set, or the set containing no elements. Note that { } 2. Set Builder notation Characterize all those elements in the set by stating the property or properties they must have to be members. D = {x x is prime and x > 2} E = {x x is odd and x > 2} The vertical bar means such that N = {0, 1, 2, 3, } is the set of natural numbers Z = {, -2, -1, 0, 1, 2, } is the set of integers Z + = {1, 2, 3, } is the set of positive integers Z - = {-1, -2, -3, } is the set of negative integers Q = {p/q : p, q Z, q 0} is the set of rational numbers Any number that can be expressed as a fraction of two integers (where the bottom one is not zero) R is the set of real numbers R +, the set of positive real numbers C, the set of complex numbers. 2
x S means x is an element of set S. x S means x is not an element of set S. Example: 4 {1, 2, 3, 4} 7 {1, 2, 3, 4} Universal set U : the set of all of elements (or the universe ) from which given any set is drawn. For the set {-2, 0.4, 2}, U would be the real numbers For the set {0, 1, 2}, U could be the N, Z, Q, R depending on the context For the set of the vowels of the alphabet, U would be all the letters of the alphabet Venn diagrams Sets can be represented graphically using Venn diagrams universal set U, which contains all the objects under consideration, is represented by a rectangle. Circles or other geometrical figures Inside this rectangle are used to represent sets. Points represent the particular elements of the set. S U 3
Sets can contain other sets S = { {1}, {2}, {3} } T = { {1}, {{2}}, {{{3}}} } V = { {{1}, {{2}}}, {{{3}}}, { {1}, {{2}}, {{{3}}} } } V has only 3 elements! Note that 1 {1} {{1}} {{{1}}} They are all different { } The first is a set of zero elements The second is a set of 1 element Replace by { }, and you get: { } {{ }} It s easier to see that they are not equal that way Definition: Two sets are equal if and only if they have the same elements. Therefore, 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. Definition: The set A is a subset of B if and only if every element of A is also an element of B.We use the notation A B to indicate that A is a subset of the set B. Examples: {1, 2, 3, 4, 5} = {5, 4, 3, 2, 1} but {1, 2, 3, 4, 5} {1, 2, 3, 4} If A = {2, 4, 6}, B = {1, 2, 3, 4, 5, 6, 7}, A is a subset of B This is specified by A B meaning that x (x A x B) 4
A B means A is a subset of B. or, B contains A. or, every element of A is also in B. or, x ((x A) (x B)). For any set S, S S ( S S S) For any set S, S ( S S) Every non-empty set S, it has at least two subsets: 1) S A B 2) S S U Venn Diagram A B means A is a subset of B. A B means A is a superset of B. A = B if and only if A and B have exactly the same elements. iff, A B and B A iff, A B and A B iff, x ((x A) (x B)). 5
Proper Subsets A B means A is a proper subset of B. A B, and A B. x ((x A) (x B)) x ((x B) (x A)) A B U The difference between subset and proper subset is like the difference between less than or equal to and less than for numbers. {1,2,3} {1,2,3,4,5} {1,2,3} {1,2,3,4,5} Examples: Let B = {0, 1, 2, 3, 4, 5} If A = {1, 2, 3}, A is not equal to B, and A is a subset of B A proper subset is written as A B Sets may contain other sets as members: A = {0, {a}, {b}, {a, b}} and B = {x I x is a subset of the set {a, b}}. Note that A = B, {a} A, but a A 6
Quick examples: Is {1,2,3}? Yes! x (x ) (x {1,2,3}) holds, because (x ) is false. Is {1,2,3}? NO Is {1,2,3}? Yes Is {,1,2,3}? Yes Is {,1,2,3}? Yes Is {x} {x}? No Cardinality of Set If S is finite, then the cardinality of S is the number of distinct elements in S. The cardinality of S is denoted by S. Examples: If S = {1, 2, 3, 4, 5}, Then S = 5 If S = {3,3,3,3,3}, Then S = 1 If S =, Then S = 0 If S = {, {a}, {b}, {a, b}}, Then S = 4 If S be the set of odd positive integers less than 10 Then S = 5. If S be the set of letters in the English alphabet Then S = 26 7
Power Sets The power set of S is the set of the subsets of S. (written as P(S)) Given S = {0, 1, 2}. All the possible subsets of S? P(S) = {, {0}, {l}, {2}, {0, 1}, {0, 2}, {l, 2}, {0, 1, 2}} Note that S = 3 and P(S) = 8 If S has n elements then the power set of S has 2 n elements If S is a set, then the power set of S is 2 S = { x : x S }. If S = {a}, then P(S) =2 S = {, {a}}. If S = {,{ }}, then P(S) = 2 S = {, { }, {{ }}, {,{ }}}. Fact: if S is finite, P(S) = 2 S = 2 S. (if S = n, 2 S = 2 n ) Cartesian Product The Cartesian Product of two sets A and B is: A x B = { < a, b > : a A b B} Example: Given A = { a, b } and B = { 0, 1 }, what is their Cartesian product? C = A x B = { (a,0), (a,1), (b,0), (b,1) } Example: What is the Cartesian product of: A = { I, 2} and B = {a, b, c} Solution: The Cartesian product A x B is: A B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c) }. Note that: A B B A Since, B A = {(a, I), (a, 2), (b, I), (b, 2), (c, 1), (c, 2)} A 1 x A 2 x x A n = {<a 1, a 2,, a n >: a 1 A 1, a 2 A 2,, a n A n } 8
Example: What is the Cartesian product A B C, where A = {0, l}, B = { I, 2}, and C = {0, 1, 2}? Solution: A B C = {(0, 1, 0), (0, I, I), (0, 1, 2), (0, 2, 0), (0, 2, I), (0, 2, 2), (1, 1, 0), (1, 1, 1), (1, 1, 2), (1, 2, 0), (1, 2, 1), (1, 2, 2)}. Exercises 1. List the members of these sets. a) {x I x is a real number such that x 2 = I} b) {x I x is a positive integer less than 12} c) {x I x is the square of an integer and x < 100} d) {x I x is an integer such that x 2 = 2} 2. Use set builder notation to give a description of each of these sets. a) {0, 3, 6, 9, I2} b) {-3, -2, -I,0, I, 2, 3} c) {m, n, o, p} 3. Determine whether each of these pairs of sets are equal. a) {I, 3, 3, 3, 5, 5, 5, 5, 5}, {5, 3, I} b) {{I}}, {I, {I}} c), { } 4. Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of these sets are subsets of which other of these sets. 9
Exercises Exercises 10
Exercises Chapter 2.2 Set Operations 11
Two sets can be combined in many different ways: Definition: Union The union of the sets A and B, denoted by A U B, is the set that contains those elements that are either in A or in B, or in both. A B = { x : x A v x B} B A Examples: {1, 2, 3} U {3, 4, 5} = {1, 2, 3, 4, 5} {a, b} U {3, 4} = {a, b, 3, 4} {1, 2} U = {1, 2} Properties of the union operation A U = A A U U = U A U A = A A U B = B U A A U (B U C) = (A U B) U C Identity law Domination law Idempotent law Commutative law Associative law 12
Intersection The intersection of the sets A and B, denoted by A B, is the set containing those elements in both A and B A B = { x : x A x B} B A Examples: {1, 2, 3} {3, 4, 5} = {3} {a, b} {3, 4} = {1, 2} = Properties of the intersection operation A U = A A = A A = A A B = B A A (B C) = (A B) C Identity law Domination law Idempotent law Commutative law Associative law 13
Disjoint Formal definition for disjoint sets: two sets are disjoint if their intersection is empty set A B = Examples: {1, 2, 3} and {3, 4, 5} are not disjoint {a, b} and {3, 4} are disjoint {1, 2} and are disjoint Their intersection is empty set and are disjoint! Their intersection is empty set Complement Let U be the universal set. The complement of the set A, denoted by A, is the complement of A with respect to U. The complement of the set A is U - A. The complement of a set A is: A = { x : x A} = A c Note that: = U and U = Properties of complement sets (A c ) c = A Complementation law A U A c = U Complement law A A c = Complement law Example: Let U is the set of all positive integers If A = {x x > 10}, Then A c = {l,2,3,4,5,6,7,8,9,l0}. A A 14
Difference The difference of A and B, denoted by A - B, is the set containing those elements that are in A but not in B. The difference of two sets A and B is: A - B = { x : x A x B } A - B = A B U Examples: {1, 2, 3} - {3, 4, 5} = {1, 2} {a, b} - {3, 4} = {a, b} {1, 2} - = {1, 2} The difference of any set S with the empty set will be the set S S - = S, where S is any set 15
symmetric difference The symmetric difference of two sets A and B is: A B = { x : (x A x B) v (x B x A)} = (A - B) U (B - A) = (A B) (A B) Set identities 16
How to prove a set identity? For example Prove that : (A U B) = A B Four methods: Use the basic set identities Use membership tables Prove each set is a subset of each other Use set builder notation and logical equivalences 1- By using the basic set identities (A U B) = A U B = A B = A B 17
2- Bu Using membership tables 3- By using each set is a subset of each other 1. ( ) (x A U B) (x A U B) (x A and x B) (x A B) 2. ( ) (x A B) (x A and x B) (x A U B) (x A U B) 18
4- By using set builder notation and logical equivalences (A U B) = {x : (x A v x B)} = {x : (x A) (x B)} = {x : (x A) (x B)} = A B Generalized Unions and Intersections Let A, B, and C be sets then: A B C contains those elements that are in at least one of the sets A, B, and C A B C contains those elements that are in all of A, B, and C. 19
Example: Let A = {0, 2, 4, 6, 8}, B = {0, 1,2,3, 4}, and C = {0, 3, 6, 9}, determine the following combinations: 1. A B C 2. A B C Solution: 1. A B C = {0, 1,2, 3,4, 6, 8, 9}. 2. A B C = {0}. Computer Representation of Sets Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The bit string (of length U = 10) that represents the set A = {1, 3, 5, 6, 9} has a one in the first, third, fifth, sixth, and ninth position, and zero elsewhere. It is 1 0 1 0 1 1 0 0 1 0. What bit strings represent the subset of: 1. All odd integers in U, A= {I, 3, 5, 7, 9}, and the bit string is 10 1010 1010. 2. Integers not exceeding 5 in U A= {l, 2, 3, 4, 5}, and the bit string is 11 1110 0000. 20
Example: The bit strings for the sets {I, 2, 3, 4, 5} and {I, 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: The bit string for the union of these sets is: 11 1110 0000 10 1010 1010 = 11 1110 1010, Corresponds to the set {l, 2, 3,4, 5, 7, 9}. The bit string for the intersection of these sets is: 11 11100000 10 1010 1010 = 10 1010 0000, Corresponds to the set {I, 3, 5}. Exercises 1. Let A = { l, 2, 3,4, 5} and B = {0, 3, 6}. Find a) A B b) A B c) A B d) B - A 2. Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find a) A B b) A B c) A B d) B A 3. Find the sets A and B if A - B = {I, 5, 7, 8}, B - A = {2, l0}, and A B = {3, 6,9} 4. Show that if A and B are sets, then a) A - B = A B c b) (A B) (A Bc)= A 5. Find the symmetric difference of: a) { I, 3, 5} and { I, 2, 3}. b) the set of computer science majors at a school and the set of mathematics majors at this school 21
6. Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3,4,5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find a) A B C b) A B C c) (A B) C d) (A B) C 7. What can you say about the sets A and B if we know that: a) A B=A b) A B=A c) A B = A d) A B= B A e) A B = B - A 8. Can you conclude that A = B if A, B, and C are sets such that: a) A C = B C b) A C = B C c) A B = B C and A C = B C 22