ELEMENTARY SET THEORY

Sets A set is a well-defined collection of objects. Each object in a set is called an element or member of the set. The elements or objects of the set are enclosed by a pair of braces { }.

Notations Capital or uppercase letters are usually used to denote sets while small or lowercase letters denote elements of a set.  denotes “is an element of” or “belongs to”  denotes “is not an element of” or “does not belong to”

Example Let A – the set of letters in the English alphabet B – the set of primary colors C – the set of positive integers g  A orange  B 100  C

Ways of Describing a Set List (or roster) method describes a set by enumerating the elements of the set. A = {a, b, c, d,…,z} B = {red, yellow, blue} C = {1, 2, 3, 4,…} Rule (or set builder) method describes a set by a statement or a rule. A = {x|x is an English alphabet} B = {a|a is a primary color} C = {y|y is a positive integer}

Definition of Terms The cardinality of a set A, or the cardinal number of A, denoted as n(A), is the number of elements in A. n(A) = 26, n(B) = 3, n(C) =  A set is finite if there is one counting number that indicates the total number of elements in the set. A and B are finite sets. A set is infinite if in counting the elements, we never come to an end. C is an infinite set.

Definition of Terms The null set or empty set, denoted by the symbol , is the set that contains no elements, that is, A is empty iff n(A) = 0. D = {x|x is a month in the Gregorian calendar with less than 28 days} n(D) = 0, so D =  A singleton set is a set that contains only one element, that is, B is a singleton set iff n(B) = 1. E = {y|y is prime number, 5 < y < 10}

Definition of Terms Sets A and B are equal if they have the same elements. Set A is a subset of B, denoted as A  B, iff every element of A is also an element of B. Laws of subset: Every set is a subset of itself, that is, A  A, for any set A. The null set is a subset of any set, that is,   A, for any set A.

Definition of Terms Example: Subset G = {x|x is an integer} F = {y|y is a whole number} C = {z|z is a positive integer} C  G because every element of C is found in G. F  C because 0F but 0C.

Definition of Terms Set A is a proper subset of B, denoted AB, if A is a subset of B and there is at least one element of B that is not in A. That is, AB iff AB and AB. P = {1, 3, 5, 7} Q = {3, 7} Then, QP The set containing all of the elements for any particular discussion is called the universal set, denoted as U.

Exercise Describe the following sets using the list method and give the set cardinality: A = {x|x is a natural number which is 1 less than a multiple of 3} B = {a|a is a rational number whose value is 2/3} C = {b|b is a vowel that appears in the phrase “set of vowels”} D = {z|z is an even prime integer greater than 2}

Exercise Describe the following sets using the rule method:

Exercise Determine which of the following statements are true and which are false:   N {1, 2, 3}  N {0}  N {1, 2, 3}  {1, 2, 3} 1  {1, 2, 3} n(N) = 1010 A  B  n(A) > n(B) {3}  N

The Power Set Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S) or (S). Example: S = {0, 1, 2} (S) = {, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}} Note that the empty set and the set itself are members of this set of subsets.

The Power Set If a set has n elements, then its power set has 2n elements. Exercise: What is the power set of A = {x, y, z}? What is the power set of the null set? What is the power set of the power set of the null set? What is the power set of B = {0, {1}, 3, {2, 4}}

SET OPERATIONS

Union Let A and B be sets. The union of the sets A and B, denoted by A  B, is the set that contains those elements that are either in A or in B, or in both. A  B = {x|x  A  x  B} Example: A = {1, 3, 5} and B = {1, 2, 3} A  B = {1, 2, 3, 5}

Union Venn Diagram

Intersection Let A and B be sets. 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} Example: A = {1, 3, 5} and B = {1, 2, 3} A  B = {1, 3}

Intersection Venn Diagram

Disjoint Two sets are called disjoint if their intersection is the empty set. A  B = { } Example: A = {1, 3, 5, 7, 9} B = {2, 4, 6, 8, 10} A  B = , then A and B are disjoint

Difference Let A and B be sets. 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 A and B is also called the complement of B with respect to A. A – B = {x|x  A  x  B} Example: A = {1, 3, 5} and B = {1, 2, 3} A – B = {5}

Difference Venn Diagram

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. In other words, the complement of set A is U – A. A’ = {x|x  A} Example: A = {a, e, i, o, u} where the universal set is the set of letters in the English alphabet A’ = {y|y is a consonant}

Complement Venn Diagram

Cartesian Products The order of elements in a collection is often important. Since sets are unordered, a different structure is needed to represent ordered collections. This is provided by 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 iff each corresponding pair of their elements is equal.

Cartesian Products Let A and B be sets. The Cartesian product of A and B, denoted by A x B, is the set of all ordered pairs (a, b) where a  A and b  B. A x B = {(a, b)|a  A  b  B} Example: A = {1, 2} and B = {a, b, c} A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)} Note: A x B  B x A

Cartesian Products The Cartesian product of sets A1, A2,…, An, denoted by A1 x A2 x … x An is the set of ordered n-tuples (a1, a2,…, an), where ai belongs to Ai for i = 1, 2,…, n. A1 x A2 x … x An = {(a1, a2, …, an) | ai  Ai for i = 1, 2, …, n} Example: A = {0, 1}, B = {1, 2}, C = {0, 1, 2}, find A x B x C.

Exercise Let A be the set of students who live one km from school and let B be the set of students who walk to classes. Describe the students in each of these sets. A  B A  B A – B B – A

Exercise Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find A  B A  B A – B B – A Let A = {0, 2, 4, 6, 8, 19}, B = {0, 1, 2, 3, 4, 5, 6} and C = {4, 5, 6, 7, 8, 9, 10} a. A  B  C b. A  B  C c. (A  B)  C d. (A  B)  C

Properties of Set Union For any sets A, B, and C, The union of any set with the null set is the set itself. A   = A The union of any set with itself is the set itself. A  A = A Set Union is commutative. A  B = B  A Set Union is associative. (A  B)  C = A  (B  C) Any set is a subset of its union with another set. A  A  B

Properties of Set Intersection For any sets A, B, and C, The intersection of any set with the null set is the null set. A   =  The intersection of any set with itself is the set itself. A  A = A Set Intersection is commutative. A  B = B  A Set Intersection is associative. (A  B)  C = A  (B  C) The intersection of any given set with another set is a subset of the given set. A  B  A A  (B  C) = (A  B) (A  C) A  (B  C) = (A  B) (A  C)

Properties of Set Difference For any sets A, B, and C, The removal of the null set from any set has no effect on the set. A –  = A The removal of the elements of any set from itself will leave the empty set. A – A =  No elements can be removed from the null set.  – A =  The result of removing the elements of a set from any given set is a subset of the given set. A – B  A A – (B  C) = (A – B)  (A – C) A – (B  C) = (A – B)  (A – C)

Properties of Set Cardinality For any sets A, B, and C, |A  B| = |A| – |A – B| |A  B| = |A| + |B| – |A  B| |A – B| = |A| – |A  B|