F.Y. B.C.A. sem. 1 Maths Ch. 1 Set theory Set : A set is well defined collection of distinct objects. Representation of Sets Sets can be represented in two ways: 1)Roster Form or Tabular form In roster form, all the elements of the set are listed, separated by commas and enclosed between curly braces { }. Example: If set represents all the leap years between the year 1995 and 2015, then it would be described using Roster form as: A ={1996,2000,2004,2008,2012} 2)Set Builder Form In set builder form, all the elements have a common property. Example: If set S has all the elements which are even prime numbers, it is represented as: S={ x: x is an even prime number} where ‘x’ is a symbolic representation that is used to describe the element. ‘:’ means ‘such that’ ‘{}’ means ‘the set of all’ So, S = { x:x is an even prime number } is read as ‘the set of all x such that x is an even prime number’. Types of sets ; 1) Finite set : Finite sets have a countable number of elements. For example, { a,b,c,d,e} is a set of five elements, thus it is a finite set. 2) Infinite set : Infinite sets contain an uncountable number of elements. For example, {1,2,3, …} is a set with an infinite number of elements, thus it is an infinite set. 3) Singleton set : It has one only element. Ex. {1} 4) Empty or null set : The empty set is a set containing no objects. It is written as a pair of curly braces with nothing inside {} or by using the symbol ∅. 5) Equal sets : Two sets are equal if they have same elements. Ex. S = { 1,2,3 } T = { 3,2,1 } Here S and T sets are equal sets. 6) Equivalent sets : Two sets are equivalent if they have same number of elements. Ex. S = { 1,2,3 } T = { a,b,c } 7) Subsets of a set : If all the elements of a set A are the elements of a set B then A is said to be a subset of B. Symbolically we write , A ⊆ B ( A is subset of B ) Ex. A = { 1,2,3 } B = { 1,2,3,4,5 } then , A ⊆ B 8) Proper sub sets : If all the elements of a set are the elements of a set B and at least one element of B is not an element of set A , then set A is called proper subset of B. Symbolically we write , A ⊂ B Ex. . A = { 1,2,3 } B = { 1,2,3,4 } then A ⊂ B 9) Power set : The family of all the subsets of a given set is called its power set. Ex. A = { 1,2 } P(A) = { ∅ , {1} , {2} , { 1,2 } } 10) Universal set : A parent set from which all different subsets are considered is know as an universal set for particular situation. Ex. A = { 1,2,3,4,……….} B = { 2,4,6,………..} Union of sets : The union of two sets S and T is the collection of all objects that are in either set. It is written S ∪ T . Using curly brace notion S ∪ T = {x : (x ∈ S) or (x ∈ T )} Suppose S = {1, 2, 3}, T = {1, 3, 5}, and U = {2, 3, 4, 5}. Then: S ∪ T = {1, 2, 3, 5}, S ∪ U = {1, 2, 3, 4, 5}, and T ∪ U = {1, 2, 3, 4, 5} Intersection of sets : The intersection of two sets S and T is the collection of all objects that are in both sets. It is written S ∩ T . Using curly brace notation S ∩ T = {x : (x ∈ S) and (x ∈ T )} Suppose S = {1, 2, 3, 5}, T = {1, 3, 4, 5}, and U = {2, 3, 4, 5}. Then S ∩ T = {1, 3, 5}, S ∩ U = {2, 3, 5}, and T ∩ U = {3, 4, 5}