NOTES ON SET THEORY AND PROPERTIES OF REAL NUMBERS DR M Shakil Mathematics Program Miami Dade College North Campus Hialeah Center 1 DEFINITION OF A SET A set is a well defined collection of distinct objects called the elements or members of the set A set is well defined if it is possible to determine whether or not a given element belongs to it NOTE Sets can be finite or infinite A set is called finite if it consists of n different elements where n is some natural number i e positive integer A set which is not finite is said to be infinite 2 DESCRIPTION OF A SET A set may be described by a listing its elements in a Roster for example A a e i o u X 0 8 2 3 7 10 etc NOTE When the list is extensive and the pattern is obvious an ellipsis may be used for example B a b c z NOTE The above form is also called the Tabular Form of the Set The elements of a set are separated by commas and are enclosed in brackets b giving a description of the elements that belong in the set for example A all vowels in the English alphabet and B all letters in the English alphabet c using set builder notation for example S x 0 x 10 and x N where N is the set of all natural numbers 3 DIFFERENT TYPES OF SETS a EQUAL IDENTICAL SETS have exactly the same elements b The UNIVERSAL SET U is the set that contains all possible elements under consideration in a problem d SUBSET A is a subset of B if every element in set A is also in set B The notation is A B NOTE Every set is a subset of itself i e A A where A is any set e PROPER SUBSET A is a proper subset of B notation A B if every element in set A is also in set B and at least one element in set B is not in set A In other words A B if and only if A B and A B f NULL OR EMPTY SET The null or empty set is the set with no elements The notation is or NOTE The empty set is a subset of every set i e A where A is any set g NUMBER OF SUBSETS OF A SET If a set contains N elements then the number of subsets is equal to 2N and the number of proper subsets is equal to 2N 1 h POWER SET The set of all the distinct subsets of a set A is called the power set of A The notation is P A i CARDINALITY OR CARDINAL NUMBER OF A SET The number of elements in a given finite set A is called its cardinality or cardinal number The notation is n A or A 2 j NOTE If n A N then n P A 2N where A is any finite set k EQUIVALENT SETS have the same cardinality i e same number of distinct elements Thus if A and B are any two finite sets with the same number of distinct elements then n A n B 4 SET OPERATIONS a UNION OF SETS The union of sets A and B is the set of all elements which are either in A or in B The notation is A B Thus A B x x A or x B or x both A and B b INTERSECTION OF SETS The intersection of sets A and B is the set of all those elements which are common to both A and B The notation is A B Thus A B x x A and x B c DIFFERENCE OF TWO SETS Let A and B be two sets Then the difference of sets A and B denoted by A B is the set of all those elements of A which do not belong to B Thus A B x x A and x B Similarly the difference of sets B and A denoted by B A is the set of all those elements of B which are not in A i e B A x x B and x A d COMPLEMENT OF A SET Let A be a set and U the universal set Then the complement of set A with respect to the universal set U is the difference of sets U and A i e U A and is denoted by A or A Thus A x x U and x A e DISJOINT SETS Any two sets A and B are said to be disjoint if and only if A B 5 NOTATIONS USED WITH SETS i is an element of ii is not an element of iii A B A is a subset of B iv A B A is a proper subset of B v A B A is not a subset of B vi U Universal set vii Empty set viii A Complement of set A ix A B A union B x A B A intersection B xi or means such that 3 6 VENN DIAGRAMS i Geometric pictures are extremely helpful but not necessary for mathematical thinking and discussions In particular we can visualize sets and operations on sets and guess the truth of a number propositions on sets with the help of geometric diagrams known as Venn diagrams These diagrams are named in honor of the English mathematician John Venn 1831 1923 who invented these and used them to illustrate ideas in his text on symbolic logic published in 1881 ii A convenient way to use Venn Diagrams is to represent the universal set U by rectangular area in a plane and the elements which make up U by the points of this area Sets can be visualized as parts of the rectangular area In particular simple plane areas bounded by circles or their parts drawn within the rectangular area can give a simple and instructive picture of sets and operations on sets We can think of each set as consisting of all points within the corresponding circle iii A subset A of set B is represented by drawing the circle inside the circle representing the set B A single circle is used to represent equal sets Two circles drawn in such a way that they have no common area represent two disjoint sets Two sets having common elements are represented by closed figures generally circles having some area common to both EXAMPLES ON SET OPERATIONS Let U 2 3 4 5 7 9 denote the universal set in a problem under consideration and let X 2 3 4 5 Y 3 5 7 9 and Z 2 4 5 7 9 be any three sets of the said problem List the members of each of the following sets using set braces Draw Venn Diagrams also i X Z ii X Y Z AN EXAMPLE ON APPLICATION OF VENN DIAGRAM A survey of 53 business executives shows that 10 play golf and tennis 9 play all three 25 play bridge …