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### SetTheory(Dr. Praveen Mittal).pdf

• 1. Discrete Mathematics BCSC1010 Module 1 Dr. Praveen Mittal Sets(Lecture1) Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 2. Introduction Discrete mathematics is the study of discrete objects Discrete means ‘distinct or not connected’ Modern mathematics deals with sets not numbers Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 3. Sets A set is an unordered collection of objects The objects in a set have similar properties The objects in a set are called the elements or members of the set Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 4. Sets Lowercase letters are usually used to denote elements of sets a,b,c,….,x,y,z Sets are denoted by uppercase letters A,B,C,…..,X,Y,Z x ϵ A denotes that ‘x’ is an element of the set A x A denotes that ‘x’ is not an element of the set A Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 5. Some Standard Sets N = {1,2,3, ... }, the set of natural numbers Z = { ... , -2, -1,0, 1,2, ... }, the set of integers Z+ = {I, 2, 3, ... }, the set of positive integers Q = {p/q | p ϵ Z, q ϵ Z, and q ≠0}, the set of rational numbers R= the set of real numbers C=the set of all complex numbers Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 6. Representation of Sets Roaster or Tabular form Rule Method or Set Builder form Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 7. Roaster or Tabular form The notation {a, b, c, d} represents the set with the four elements a, b, c, and d Example: The set V of all vowels in the English alphabet can be written as V = {a, e, i, o, u} Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 8. Roaster or Tabular form (Examples) The set O of odd positive integers less than 10 can be expressed by O= {1, 3, 5, 7, 9} The set of positive integers less than 100 can be denoted by {1, 2, 3, ... , 99} Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 9. Rule Method or Set Builder form We can also "build" a set by describing what is in it  It says "the set of all x's, such that x is greater than 0".  In other words any value greater than 0  Sometimes ":" can be used instead of "|", so we can write { x : x > 0 } Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 10. Rule Method or Set Builder form Ref: https://www.mathsisfun.com/sets/set-builder-notation.html Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 11. Rule Method or Set Builder form Ref: https://www.mathsisfun.com/sets/set-builder-notation.html Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 12. Types of Set Finite Set Infinite Set Null Set Singleton Set Subset Super Set Proper Subset Equal Set Universal Set Power Set Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 13. Empty Set  A set with no elements is called the null or empty set. It is represented by the symbol { } or Ø . Examples  the set of months with 32 days  The set of dogs with six legs  The set of squares with 5 sides  The set of cars with 20 doors  The set of integers which are both even and odd Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 14. Empty Set (Example) A={x: x is natural number less than 1} Since there is no such natural number exists. Thus, A={} or Ø Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 15. Finite Set Finite sets are the sets having a finite/countable number of elements. Examples A set of English alphabets; E={a,b,c,…..,z} A set of natural numbers less than 6; A={1,2,3,4,5} Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 16. Infinite Set If the number of elements in a set is not countable/infinite, then it is called an infinite set. Examples  Set of all natural numbers {1,2,3,…}  A set of all points on a line  The set of leaves on a tree Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 17. Singleton Set A set which contains only one element is called a singleton set. For example: B = {x : x is a whole number, x < 1} This set contains only one element 0 and is a singleton set. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 18. Singleton Set  Let A = {x : x N and x² = 4} Here A is a singleton set because there is only one element 2 whose square is 4.  Let B = {x : x is an even prime number} Here B is a singleton set because there is only one prime number which is even, i.e., 2. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 19. Subset  If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B and we write it as A B  Symbol ‘ ’ is used to denote ‘is a subset of’ or ‘is contained in’.  Every set is a subset of itself  Null set or is a subset of every set.  Number of subsets of a set= where n is total number of elements in a set Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 20. Subset (Examples)  Let A = {2, 4, 6} B = {6, 4, 8, 2} Here A is a subset of B, since all the elements of set A are contained in set B.  Vowels in English  Population of Mathura in Uttar Pradesh Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 21. Super Set  Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B A.  Symbol is used to denote ‘is a super set of’  Example A = {a, e, i, o, u} B = {a, b, c, ............., z} Here A B i.e., A is a subset of B but B A i.e., B is a super set of A Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 22. Proper Subset If A and B are two sets, then A is called the proper subset of B if A B but A ≠ B It is denoted by A B The symbol ‘ ’ is used to denote proper subset No set is a proper subset of itself Null set or is a proper subset of every set Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 23. Proper Subset (Example) A = {1, 2, 3, 4} B = {1, 2, 3, 4, 5} We observe that, all the elements of A are present in B but the element ‘5’ of B is not present in A. So, we say that A is a proper subset of B. Symbolically, we write it as A B Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 24. Equal set Two sets are equal if and only if they have the same elements That is, if A and B are sets, then A and B are equal if and only if x(x ϵ A ↔ x ϵ B). It is denoted by A = B. If A B and B A then A=B Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 25. Equal set (Examples) The sets {1, 3, 5} and {3, 5, 1} are equal because they have the same elements Set {1, 3, 3, 3, 5, 5, 5, 5} is same as the set {1, 3, 5} because they have the same elements. Order of elements is meaningless It does not matter how often the same element is listed. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 26. Universal Set A set which contains all the elements of other given sets is called a universal set. The symbol for denoting a universal set is . Example:  If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7} then U = {1, 2, 3, 4, 5, 7} [Here A U, B U, C U and U A, U B, U C] Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 27. Power Set  The collection of all subsets of set A is called the power set of set A.  It is denoted by P(A). Example If A = {p, q} then all the subsets of A will be P(A) = { , {p}, {q}, {p, q}} Number of elements of P(A) = n[P(A)] = where m is the number of elements in set A. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 28. Find the power set of {0,1,2} set Solution:  The power set P({0, I, 2}) is the set of all subsets of {0, 1, 2}. Hence, P({0,1,2})={Φ,{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. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 29. Cardinality of a Set Cardinality of a set S, denoted by |S| is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞. Example − | {1,4,3,5} | = 4, | {1,2,3,4,5,…} | = ∞ Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
• 30. In next lecture we will discuss… Operations on sets Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
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