Chapter two Theory of sets Afjal hossain Assistant professor Department of marketing, Pstu

Theory of sets term Definition with examples Set A set is a collection of well-defined and well-distinguished objects. From a set it is possible to identify whether a given object or element belongs to a set or not. Ex: A = {2, 3, 4} A = {all even integers} etc. Elements of a Set The objects that make up a set are called the elements of a set. It is also known as members of a set or belongs to a set. Ex: A = {2, 3, 4}; Here 2, 3, 4 are the elements of Set A. In other words, 2  A.

Types of sets term Definition with examples Finite Set When the elements of a set can be counted by a finite or specific number of elements, then it is called a finite set. Symbolized as F. Ex: A = {2, 5, 8} Infinite Set When the total number of elements of a set cannot be counted then it is called infinite set. Symbolized as F∞. Ex: A = {1, 2, ………..}

Types of sets term Definition with examples Singleton Set Also known as Unit Set. A set containing only one element is called a singleton set. Symbolized as S. Ex: A = {5} Empty Set Also known as Null/ Void Set. A set having no element is called an empty set. The symbol used to an empty set is a Greek letter ø. Ex: A = { ø}

Types of sets term Definition with examples Equal Set Two sets A & B are said to be equal whenever every elements of Set A is also an element of Set B and vice versa, then it becomes an equal set. In other word, Two set having same elements are called equal sets. It is also known as “Axiom of Extension or Axiom of Identity” Symbol: A =B. Ex: if A = {2, 3} and B = {3, 2} Equivalent Set If the total numbers of elements of one set are equal to the total number of elements of another set, then two sets are equivalent. The total number of elements may be equal but the elements of the two sets may not be the same. In other word, two sets having same number of distinct elements are called equivalent set. Symbol: . Ex: if A = (a, b, c} and B = {1, 2, 3}

Types of sets term Definition with examples Subset If every element of set A is also an element of set B then A is called the subset of B. And B is called the super set of A. Symbol:  Ex: A = {2, 3} and B = {1, 2, 3, 4} then AB. Proper Subset If all the elements of set A are also the elements of Set B but at least one element of Set B is not an element of Set A, then A is called a proper subset of B. Symbol:  Ex: A = {1, 2, 5} and B = {1, 2, 4, 5} and B is the super proper subset of A.

Types of sets term Definition with examples Power set The family of all subsets of a set is called the power set of that set and is denoted as P(S). Ex: if A = {1, 2} then P(S) = { ø, {1}, {2}, {1, 2}} Universal set A set containing all the elements is called a Universal set. In other word, if all the sets are considered as subsets of a fixed set, then that fixed set is called the universal set. This set is usually denoted as U. Ex: if A = {1, 2} and B = {4} then U = {1, 2, 4} Venn Diagram Pictorial presentation of set theory is called Venn Diagram. It was named in line with the name of English Logician John Venn. Generally the universal set is enclosed by a rectangle and one or more sets are shown throw circle with that rectangle.

intersection of sets The intersection of two sets A and B is the set consisting of all the elements that belong to both A and B. The intersection of A and B is denoted as A ∩ B. Ex: if A = {2, 5, 7} and B = {4, 5, 8} then A ∩ B = {5}  Properties A intersection B is the subset of both A & B i.e. A ∩ B  A/B Intersection of any set with an empty set is a null set i.e. A ∩  =  Intersection of any set with itself is the set itself i.e. A ∩ A=A It has a commutative property i.e. A ∩ B = B ∩ A Intersection has an associative property i.e. A ∩ (B ∩ C)=(A ∩ B) ∩ C If A is a subset of B then B intersection A is equal to A and if B is a subset of A then A intersection B equals B If A is a subset of B and B is the subset of C then A is the subset of B intersection C

Union of sets The Union of two sets A and B is the set consisting of all the elements that belong to either A or B or both. The Union of two sets A and B is also called the Logical Sum of A and B or A cup B or A union B. Union is also known as “Join” or “Logical Sum” of A and B. Symbol: U. Ex: A = {2, 3} and B = {1, 4} then A U B = {1, 2, 3, 4} Properties The individual sets composing a Union are the members or subsets of that union Union of a set with itself is the set itself i.e. A U A = A Union of a set with an empty set is a set itself i.e. A U  = A It has a commutative property i.e. A U B = B U A It has an associative property i.e. A U (B U C) = (A U B) U C B is a subset of A, then A union B equals A and A is a subset of B then A union of B equals B A U B = {ø} if A = {ø} and B = {ø}

Types of sets term Definition with examples Distribution Laws of Union & Intersection A U (B ∩ C) = (A U B) ∩ (A U C) A ∩ (B U C) = (A ∩ B) U (A ∩ C) Compliment of a Set The complement of a set is the set of all those elements which do not belong to that set. It is defined with as respect to the Universal Set “U”. If “U” is an Universal Set and “A” is a subset of U, then the complement of Set A is the Set (U-A) and is denoted as Ấ. Properties The intersection of a set and its complement is a null set The union of a set and it’s complement is the universal set The complement of the complement of a set is the set itself If A is a proper subset of B then B complement is a proper subset of A complement

Types of sets term Definition with examples De-Morgan’s Law The complement of a Union is the Intersection of the complement The complement of an Intersection is the Union of the complement Difference of Two Sets The difference of two sets A and B is the set of all those elements which belong to A but not B. the difference of A and B is denoted by A-B or A B and read as A difference B or A minus B. And the difference of two sets B and A is the set of all those elements which belong to B but not to A and symbolically expressed as B-A. Properties (A-B) is the subset of A and (B-A) is the subset of B (A-B), (A ∩ B) and (B-A) are mutually disjoints A-(A-B) = A ∩ B and B-(B-A) = A ∩ B (A-B) ∩ B = ø

Just Remember !!! U = Or, only, neither--------nor, anyone, not all of these ∩ = and, both, but, all, either------or, only one, first not second or third. Formula: n(A) = n(A∩B) + n(A∩B́) n(A∩B) = n(A) - n(A∩B́) n(A U B) = n(A) + n(B) - n(A∩B) n(A U B)́ = n(U) - n(A U B) n(A U B U C) = n(A) + n(B) + n(C) - n(A∩B) – n(A∩C) - n(B∩C) + n(A∩B∩C) n(A U B U C)́ = n(U) – n(A U B U C) n(Á) = n(U) - n(A) n(A∩B∩C′) = n(A∩B) - n(A∩B∩C) n(A∩B′∩C′) = n(A) - n(A∩B) - n(A∩C) + n(A∩B∩C)

Exercise ! A survey of 400 students in a university shows that 100 are smokers, 150 as chewers of gum, 75 are both smokers and gum chewers. Find out how many students are neither smokers nor gum chewers. A student surveyed 525 people. He determined that 350 read newspaper, 215 listened to radio and 140 watched television. Additionally 75 read the newspaper and watched television, 40 listened radio and watched television and 100 read the newspaper and listened to radio. If 25 used all three sources of news, how many people utilized none of the three. A student surveyed 525 people. He determined that 350 read newspaper, 315 listened to radio and 140 watched television. Additionally 75 read the newspaper and watched television, 40 listened radio and watched television and 100 read the newspaper and listened to radio. If 25 used all three sources of news, are the information correct?

Exercise ! A survey of 400 students in a university shows that 100 are smokers, 150 as chewers of gum. Find out how many students are both of smokers and gum chewers. From the statistics of 45 students of a class, it was revealed that 25 students took mathematics, 21 took economics and 15 took law as additional courses. Of them 12 took mathematics and economics, 9 took mathematics and law and 7 took economics and law, 3 took mathematics, economics and law. How many took only mathematics but not economics and law? How many did not take any additional course at all?

Exercise ! There are 1,500 students who appeared in CMA examination under the ICMAB. Out of these students, 450 failed in Accounting, 500 failed in Business Mathematics and 475 failed in Costing. Those who failed in Accounting and Business Mathematics were 300, those who failed in Business Mathematics and Costing were 320 and those who failed in both Accounting and Costing were 350. The students who failed in all the three subjects were 250. How many students who failed at least in any one of the subjects? How many students who failed in no subjects? How many students who failed in only one subject? How many students who failed in only Accounting and Business Mathematics?