Denoting the beginning SET THEORY What is a set? A set is a collection of distinct objects. The objects in a set are called the elements or the members of the set. The name of the set is written in upper case and the elements of the set are written in lower case. If x is an element of a set A, we say that x belongs to or is a member of A, and is expressed symbolically as x A. If y is not a member of A, then this is symbolically denoted as y A Let V be the set of all vowels. Then V is written as V= {a, e, i, o, u} Name of the set The curly brackets Denoting the beginning And end The definition of a set should be such that given an element, you can unambiguously judge whether that Kavita Hatwal Fall 2002

The order in which elements appear do not matter, so {a,o,i,e, u}, {u, e, o, a, i}, {o, e, a, i,u} are all V a {a} a is an element whereas {a} is a set whose element is a. {} makes all the difference in set notation. Page 242, #6-c, d Page 266, #6-c, d Sets given by defining property. {x R | -2 < x < 5 } Is read as (from left to right) the set of all x such that x is a real number and also x is greater than –2 and less than 5. Can you list some of the members of this set? Page 242, #4-b Page 266, #4-b Finite and infinite sets. Sets whose elements can be listed are called finite sets, like H={seasons in a year} Which of the above sets are finite and which are infinite? Kavita Hatwal Fall 2002

Can you find any x which satisfies the above condition? SUBSETS If A and B are sets , A is called the subset of B, written as A B, if and only if, every element of A is also an element of B. Symbolically, A B , x, if x A then x B. Also A is contained in B or B contains A are ways of saying that A is a subset of B. Page 242, #6-g, h Page 266, #6-g, h EMPTY SET Consider the set X= Can you find any x which satisfies the above condition? A = {set of all animals} P={x A| x is a pink elephant} What could possibly be the elements of P? A set with no elements is called an empty set denoted as . An empty Set is a subset of every set. A, where A is any set. Kavita Hatwal Fall 2002

Is in B but there is at least one element of B that is not in A because is the set with no elements {}, whereas { } is the set with one element, the empty set. Subsets revisited Let A and B be sets. A is a proper subset of B if, and only if, every element of A Is in B but there is at least one element of B that is not in A For every set A A A, i.e. each set is its own subset And A, i.e. the empty set is subset of every set U is the universal set or universe of discourse. It is considered the all encompassing set. Every set is a subset of U. Kavita Hatwal Fall 2002

For example, if A and B are sets and Venn Diagrams What is a Venn Diagram? A Venn Diagram is a pictorial representation of sets. U is the universal set which is represented as a rectangle. Other sets are represented as circles. For example, if A and B are sets and A B, that is B contains A, then this situation is represented as More on Venn Diagrams later Z = set of integers Q = set of rational numbers. R= set of real numbers Can you show the subset relation between Z, Q and R using notation and Venn Diagram notation ? A B Kavita Hatwal Fall 2002

element of A is in B and every element of B is in A. Symbolically, Set Equality Venn Diagram Page 243, #10-a, c Page 266, #10-a, c Given two sets A and B, A equals B, written A = B, if, and only if, every element of A is in B and every element of B is in A. Symbolically, A=B Kavita Hatwal Fall 2002

Let A and B be subsets of the universal set U. Set Operations. Let A and B be subsets of the universal set U. The Union of A and B , denoted , is the set of all elements x in U such that x is either in A or in B. 2. The Intersection of A and B , denoted , is the set of all elements x in U such that x is in both A and B. 3. The Difference of B minus A , denoted B-A, is the set of all elements x in U such that x is in B, but not in A. The complement of a set A, denoted as is the set of all elements x in U such that x is not in A. Symbolically Kavita Hatwal Fall 2002

Show the facts of previous slides using Venn Diagrams Class Activity Draw the rest on your own Page 242, #7 Page 266, #7 Kavita Hatwal Fall 2002

A  B = B  A. A  B = B  A. c = U. Uc = . Complement A  Ac = . A  Ac = U. Distributivity A  (B  C) = (A  B)  (A  C). A  (B  C) = (A  B)  (A  C). Identity A  U = A. A   = A. Commutativity A  B = B  A. A  B = B  A. Kavita Hatwal Fall 2002

Associativity (A  B)  C = A  (B  C). (A  B)  C = A  (B  C). Idempotent A  A = A. A  A = A. Universal Bounds A   = . A  U = U. Two sets A and B are called disjoint if they have no elements in common, i.e. A  B =  Page 267, #6-a, 8 Page 268, #19-a, 290 #4 Kavita Hatwal Fall 2002

Given 2 sets A and B, the Cartesian Product of A and B , denoted , is the set of all ordered pairs(a,b), where a is in A and b is in B. Given sets the Cartesian Product of denoted by is the set of all ordered n-tuples where Symbolically, Page 238, example 5.1.10 Page 265, example 5.1.15 Given a set A, the power set of A, denoted P (A), is the set of all subsets of A. For all sets A and B, then 1. 2. Page 268, #40 Page 268, #26 Kavita Hatwal Fall 2002

Two sets A and B are disjoint if A  B = . Sets A1, …, An are pairwise disjoint if Ai  Aj =  for all i, j  {1, …, n}, with i  j. A collection of sets {A1, …, An} is a partition of a set A if A1  …  An = A, and A1, …, An are pairwise disjoint. Kavita Hatwal Fall 2002

Then {A0, A1, A2} is a partition of Z. Let A0 = {n  Z | n = 3k for some k  Z}. A1 = {n  Z | n = 3k + 1 for some k  Z}. A2 = {n  Z | n = 3k + 2 for some k  Z}. Then {A0, A1, A2} is a partition of Z. Kavita Hatwal Fall 2002

Set Identities Kavita Hatwal Fall 2002

A Boolean algebra is a set S that includes two elements 0 and 1, Boolean Algebras A Boolean algebra is a set S that includes two elements 0 and 1, has two binary operations + and , has one unary operation , which satisfy the following properties for all a, b, c in S: Kavita Hatwal Fall 2002

Boolean Algebras Commutativity a + b = b + a. a  b = b  a. Associativity (a + b) + c = a + (b + c). (a  b)  c = a  (b  c). Kavita Hatwal Fall 2002

Boolean Algebras Distributivity a  (b + c) = (a  b) + (a  c) Identity a + 0 = a. a  1 = a. Kavita Hatwal Fall 2002

Boolean Algebras Complementation a + a = 1. a  a = 0. Kavita Hatwal Fall 2002

Examples: Boolean Algebras Let U be a nonempty universal set. Let 0 be  and 1 be U. Let + be  and  be . Let  be complementation. Then U is a Boolean algebra. Let U be a nonempty universal set. Let 0 be U and 1 be . Let + be  and  be . Let  be complementation. Then U is a Boolean algebra. Kavita Hatwal Fall 2002

Examples: Boolean Algebras Let S be the set of all statements. Let 0 be F and 1 be T. Let + be  and  be . Let  be negation. Let = be . Then S is a Boolean algebra. Let S be the set of all statements. Let 0 be T and 1 be F. Let + be  and  be . Let  be negation. Let = be . Then S is a Boolean algebra. Kavita Hatwal Fall 2002

Theorem: Let S be a Boolean algebra and let a, b in S. Then a  a = a. Derived Properties Theorem: Let S be a Boolean algebra and let a, b in S. Then a  a = a. a + a = a. a  0 = 0. a + 1 = 1. a  b = a if and only if a + b = b. a  b = a + b if and only if a = b. Kavita Hatwal Fall 2002

Derived Properties, continued a  b = 0 and a + b = 1 if and only if a = b. 0 = 1. 1 = 0. (a) = a. (a  b) = a + b. (a + b) = a  b. Kavita Hatwal Fall 2002

Example: Boolean Algebra Let S = {1, 2, 3, 5, 6, 10, 15, 30}. Define a + b = gcd(a, b). a  b = lcm(a, b). a = 30/a. 1 is U, 30 is 0 Verify the 10 basic properties. http://en.wikipedia.org/wiki/Greatest_common_divisor Kavita Hatwal Fall 2002