Set theoretic operation using Venn diagrams MADE BY-DEVPRIYA UJJWAL CLASS-11TH-A WHAT ARE SETS ? In mathematics, a set is a collection of distinct elements. The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets. Two sets are equal if and only if they have precisely the same elements. Sets, in mathematics, are an organized collection of objects and can be represented in set-builder form or roster form. Usually, sets are represented in curly braces {}, for example, A = {1,2,3,4} is a set. WHAT ARE VENN DIAGRAMS ? A Venn diagram is a widely-used diagram style that shows the logical relation between sets, popularized by John Venn in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. UNION OF SETS The union of two sets A and B is defined as the set of elements which are either in set A or set B or in both A and B. This operation is denoted by the symbol. ∪ INTERSECTION OF SETS In mathematics, the intersection of two sets A and B, denoted by A B, is the set containing all elements of A that also belong to B (or equivalently, all elements of B that also belong to A). he set operation intersection takes only the elements that are in both sets. The intersection contains the elements that the two sets have in common. The intersection is where the two sets overlap. ∩ COMPLEMENT OF SET If U is a universal set and A be any subset of U then the complement of A is the set of all members of the universal set U which are not the elements of A. A′ = {x : x U and x A} Alternatively it can be said that the difference of the universal set U and the subset A gives us the complement of set A. ∈ ∉ DISJOINT SET A and B are disjoint. Here, we observe that there is no common element in A and B. Therefore, n(A B) = n(A) + n(B) ∪ NOT A DISJOINT SET When A and B are not disjoint, we have from the figure (i) n(A B) = n(A) + n(B) - n(A B) (ii) n(A B) = n(A - B) + n(B - A) + n(A B) (iii) n(A) = n(A - B) + n(A B) (iv) n(B) = n(B - A) + n(A B) ∪ ∪ ∩ ∩ ∩ ∩ DIFFERENCE OF A-B Let A and B be two sets. The difference of A and B, written as A - B, is the set of all those elements of A which do not belongs to B. Thus A – B = {x : x A and x B} or A – B = {x A : x B}. Clearly, x A – B x A and x B In the adjoining figure the shaded part represents A – B. ∈ ∈ ∉ ∈ ⇒ ∈ ∉ ∉ DIFFERENCE OF B-A Similarly, the difference B – A is the set of all those elements of B that do not belongs to A. Thus, B – A = {x : x A and x B} or A – B = {x B : x A}. ∉ ∈ ∈ ∉ VENN DIAGRAM OF THREE SETS The formula used to solve the problems on Venn diagrams with three sets is given below: n(A ⋃ B ⋃ C) = n(A) + n(B) + n(C) – n(A ⋂ B) – n(B ⋂ C) – n(A ⋂ C) + n(A ⋂ B ⋂ C) THREE SETS SUBSET OF SET In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. QUESTIONS ON VENN DIAGRAM >IN A SURVEY OF SCHOOL STUDENTS, 64 HAD TAKEN MATHEMATICS COURSE, 94 HAD TAKEN CHEMISTRY COURSE, 58 HAD TAKEN PHYSICS COURSE, 28 HAD TAKEN MATHEMATICS AND PHYSICS, 26 HAD TAKEN MATHEMATICS AND CHEMISTRY, 22 HAD TAKEN CHEMISTRY AND PHYSICS COURSE, AND 14 HAD TAKEN ALL THE THREE COURSES. FIND HOW MANY HAD TAKEN ONE COURSE ONLY. From the Venn diagram above, we have- No. of students who had taken only math = 24 No. of students who had taken only chemistry = 60 No. of students who had taken only physics = 22 Total no. of students who had taken only one course := 24 + 60 + 22= 106 So, the total number of students who had taken only one course is 106. >In a school, 60 students enrolled in chemistry,40 in physics, 30 in biology, 15 in chemistry and physics,10 in physics and biology, 5 in biology and chemistry. No one enrolled in all the three. Find how many are enrolled in at least one of the subjects. From the Venn diagram, number of students enrolled in at least one of the subjects : = 40 + 15 + 15 + 15 + 5 + 10 + 0 = 100 So, the number of students enrolled in at least one of the subjects is 100. >In a school 85% of the people speak Tamil, 40% speak English and 20% speak Hindi. Also 32% speak Tamil and English, 13% speak Tamil and Hindi and 10% speak English and Hindi, find the percentage of people who can speak all the three languages. From the Venn diagram, we can have 100 = 40 + x + 32 – x + x + 13 – x + 10 – x–2+x–3+x 100 = 40 + 32 + 13 + 10 – 2 – 3 + x 100 = 95 – 5 + x 100 = 90 + x x = 100 – 90 , x = 10% So, the percentage of people who speak all the three languages is 10%. THANK YOU MADE BY – DEVPRIYA UJJWAL CLASS-11TH -A