Mathematics

A multiset is a set of elements in which every element can occur more than once. This is the different property of multiset from the sets. a set in which the occurrence of an element can be more than one or can be 0. The multiplicity of the element is the occurrence of the element in the multiset. the  multiplicity of an element can be 1 or more than one or can be 0. the size of the multiset is the sum of all the multiplicities of all the elements.  for example: A= {1,1,1,1,2,2,3,4,4,5} The multiplicity of element '1' in multiset A is 4. The multiplicity of element '2' in multiset A is 2. The multiplicity of element '3' in multiset A is 1. The multiplicity of element '4' in multiset A is 2. The multiplicity of element '5' in multiset A is 1. the size of the multiset A is 10 In sets, the cardinality of the set is the multiplicity of the set. and rest all things the representation of a set and multiset is similar. in multisets A={1,1,1,1,2,2,3,4,4...

 In this article, you will learn all about the Principle of inclusion and exclusion. the principle of inclusion and exclusion explains how to find the total number of elements that belong to the particular set. principle: for two arbitrary set A and B the principle states as : |A ᴜ B| = |A| + |B| – |A ∩ B| where |A| is the cardinality of the set A. cardinality of the set is the number of elements present in the set. for three sets the principle of inclusion and exclusion is symbolically represented as: |A ∪ B ∪ C| = |A| + |B| + |C| – | A ∩ B|  –  |A ∩ C|  –  |B ∩ C| + |A ∩ B ∩ C| Examples: example 1: Suppose a survey of 100 people asks if they have a cat or dog as a pet.  The results are as follows:  55 answered yes for the cat, 58 answered yes for dog and 20 people checked yes for  both cat and dog. How many people have a cat or a dog? solution: the people who have a cat is denoted by set C the people who have a dog is denoted by  set D |C|...

  Venn diagram is a graphical representation of sets . Or a pictorial representation of sets represented by the closed figure is called a set diagram or Venn Diagram . Venn diagrams are used to illustrate various operations like union, intersection, and difference, the complement of a set. We can explain the relationship among sets through this in a more significant way. Venn diagram consists of: A universal set by using a rectangle. A set or subset of a universal set is represented by using a circle or ovals. Examples Venn diagram for set union: Venn diagram for set intersection: Venn diagram for complement of set A: Venn diagram for the complement of set B Example:1      In a survey of students, 50 students like Mathematics, 30 people like English, 10 people likes both             English and Mathematics. find the total number of students in the survey? solution:  Given data,      Student who like Mathematic...

The operations which we can perform on the set are set intersection , set union , a  difference of two sets and complement, and the properties related to the operation  on the set . The operations on set are explained as below: Union The union of two sets A and B is a set that consist of all the elements that either belongs to set A or set B or to both. The union is denoted by '⋃' this symbol.   For example : suppose that, group A consists of five group members named, Smita, Sakshi, Monali, Akash, and Kiran, and group B consist of four members, named as  Akash, Kiran, Sanika, pakhi. The union of these two groups consists of seven members rather than 9, namely Smita, Sakshi, Monali, Akash, Kiran, Sanika, pakhi. A={1,2,3} B={3,4,5,6} A⋃B ={1,2,3,4,5,6} B⋃A ={1,2,3,4,5,6} some properties of operation of union:  A∪B = B∪A                       ( Commutative law ) A∪(B∪C) = (A∪B)∪C     ...

 The types of sets are empty set, singleton set, equivalent sets, equal sets, finite set , infinite set, subsets, super set , proper set, universal set, complement se t .     Empty set or Null set   A  set which doesnot contain any set  is called as Empty set or Null set or Void set. It is denoted by ‘ ∅’ and it is read as phi. In roster form empty set is denoted by {}. The cardinality of null set is always 0. For example: The set set of whole number less than 0. The set of even prime number less than 2.   Singleton set a set which contain only one element is called as singleton set. The cardinality of singleton set is always 1. For example: The set of even prime numbers. A = {2}, | A | = 1 A = {x|x is a whole number less than 1} S = {x|X is a natural number, x*x=4}   Finite set  A set which has a definite number of elements is called as finite set. The cardinality of finite set is always definite. For example: The...