 Hello and welcome to the session. In this session first we will discuss about Venn diagrams. Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. These diagrams consists of rectangles and closed curves Usually circles, the universal set is represented by rectangles and its subsets are represented by circles. In Venn diagrams the elements of the set are written in the respective circles. Consider this Venn diagram in which the universal set U is given by 1, 2, 3, 4, 5, 6, 7 and this circle is the set A with elements 3, 5, 6. Next we discuss operation of sets. First we have union of sets. Union of sets A and B is the set C which consists of all those elements which are either in A or in B including those which are in both. Union B is denoted by this. This is equal to X such that X belongs to A or X belongs to B. This shaded portion in this Venn diagram is the union of two sets A and B. Next we have some properties of the operations of union. First one is A union B is equal to B union A which is the commutative law then we have A union B union C is equal to A union B union C which is the associative law. Then the next one is A union phi is equal to A which is the law of identity element that is phi is the identity of union. Then next is A union A is equal to A which is the idempotent law then we have U union A is equal to U which is the law of U. Consider a set A equal to 1, 2, 3, 4 and a set B equal to 1, 2, 5 then A union B is equal to the set 1, 2, 3, 4, 5. Next we have intersection of sets. The intersection of X, A and B is the set of all those elements which belong to both A and B. Symbolically it is written as A intersection B this is the symbol that we use for intersection is equal to X such that X belongs to A and X belongs to B. This shaded portion indicates A intersection B. If A and B are two sets such that A intersection B is equal to phi then A and B are called disjoint sets. This is the Venn diagram for the disjoint sets A and B since they don't have any element common to them. Now we discuss some properties of operation of intersection. The first property is A intersection B is equal to B intersection A which is the commutative law. Then A intersection B intersection C is equal to A intersection B intersection C which is the associative law. Then we have phi intersection A is equal to phi and U intersection A is equal to A which is the law of phi and U. Then next is A intersection A is equal to A which is the idempotent law. Then we have A intersection B union C is equal to A intersection B union A intersection C which is the distributive law. Again consider set A equal to 1, 2, 3, 4 and set B equal to 1, 3, 5. A intersection B is given by 1, 3. Next we have difference of sets. The difference of sets A and B in this order is the set of elements which belong to A but not to B. Symbolically this is written as A minus B. This A minus B can be written as we set X such that X belongs to A and X does not belong to B. This shaded portion is A minus B. Consider set A equal to 1, 2, 3, 4 and a set B equal to 1, 3, 5. Now from here we have A minus B is equal to 2, 4. Next we discuss complement of a set. Let U be a universal set and A be a subset of U. Then complement of A denoted by A dash is the set of all elements of U that is X such that X belongs to U and which are not the elements of A. X does not belong to A. We have A complement is equal to universal set U minus A. If A is a subset of the universal set U then its complement that is A dash is also a subset of U. And we have A complement the whole complement is equal to A. This is the Venn diagram for the complement of the set A. This shaded portion is the A complement. Next we have some properties of complement sets. First is the complement laws which says A union A complement is equal to universal set U and A intersection A complement is equal to 5. Next we have D Morgan's law according to which we have A union B whole complement is equal to A complement intersection B complement and A intersection B whole complement is equal to A complement union B complement. The next is the law of double complementation according to which we have A complement the whole complement is equal to A. Next we have laws of empty set universal set according to which we have Phi dash that is Phi complement is equal to universal set U and universal set U dash or U complement is equal to Phi. Consider universal set U equal to 1, 2, 3, 4, 5, 6, 7, 8, 9 let's set A be equal to 2, 4, 6, 8 and set B be equal to 2, 3, 5, 7. Now from here we have A union B is equal to 2, 3, 4, 5, 6, 7, 8 then A complement is equal to 1, 3, 5, 7, 9, B complement is equal to 1, 4, 6, 8, 9. Now A union B whole complement is given by the set 1, 9 then A complement intersection B complement is the set 1, 9. So as you can see we get A union B whole complement is equal to A complement intersection B complement which is the D Morgan's law. Now we have some important formulae consider A and B be 2 finite sets. Now if we have A intersection B is equal to Phi then number of elements in the set A union B is equal to number of elements in the set A plus number of elements in the set B. If we have 2 finite sets A and B then number of elements in A union B is equal to number of elements in set A plus number of elements in set B minus number of elements in set A intersection B. The next is number of elements in the set A union B union C is equal to number of elements in the set A plus number of elements in the set B plus number of elements in the set C minus number of elements in A intersection B. Minus number of elements in B intersection C minus number of elements in A intersection C plus number of elements in A intersection B intersection C where we have A, B and C are finite sets. If suppose we have 2 sets A and B such that number of elements in the set A is 27 and number of elements in the set B is 35 and number of elements in the set A union B is 50 then let's try and find out number of elements in the set A intersection B. From this formula we get number of elements in A intersection B is equal to number of elements in A plus number of elements in B minus number of elements in A union B. So this is equal to 27 plus 35 minus 50 which comes out to be equal to 12 so we have number of elements in the set A intersection B is 12. This completes the session hope you have understood the Venn diagrams and operations on sets.