 Hello and welcome to the session. In this session, first we will discuss the algebra of sets. Let us discuss different axioms for the sets as well as for the numbers. First we have the closure axiom. In this we know the sum and product of two rational or real numbers is a rational or real number themselves. Likewise for the sets, the union and intersection two sets are themselves. Next we have the commutative axiom that is the commutative law. For numbers we have the product and the sum are commutative. That is if we consider two numbers x and y then x plus y is equal to y plus x which shows that addition is commutative and x into y is equal to y into x which shows that the product is commutative. And in case of sets also we have that intersection and union are commutative. That is a union b is equal to b union a and a intersection b is equal to b intersection a. So that union and intersection are also commutative. Next we discuss the associative property. According to the associative property for numbers we have x plus y b whole plus z is equal to x plus y plus z b whole x into y the whole into z is equal to x into y into z b whole. That is addition and multiplication obeys the associative law and for sets we have similar laws according to which we have a union b the whole union c is equal to a union b union c the whole then a intersection b the whole intersection c is equal to a intersection b intersection c the whole. Next property that we discuss is the distributive law. Now there is only one distributive law for numbers that is x into y plus z b whole is equal to x into y plus x into z and for sets we have two distributive laws according to which we have a intersection b union c the whole is equal to a intersection b the whole union a intersection c the whole and the other distributive law for sets is a union b intersection c the whole is equal to a union b the whole intersection a union c. Next we will discuss about the identities for the numbers and for the sets. For any number a we have a number 0 such that a plus 0 is equal to a also a into 1 is equal to a. In the same way for sets we have a union phi is equal to a and a intersection xi that is the universal set is equal to a. So in case of the sets the empty set phi at plus the number 0 is the union symbol corresponds to the plus sign and also this universal set xi corresponds to the number 1 if the intersection symbol corresponds to the multiplication sign. Also for any number a we have a into 0 is equal to 0 and for sets we have a intersection phi is equal to phi. So in this case this phi that is the empty set acts like the number 0 if the intersection symbol corresponds to the multiplication sign. But for any number a plus 1 would not be equal to 1 and for sets we have set a union the universal set xi is equal to the universal set xi. Next we discuss about the inverses for the numbers as well as for the sets. For any number a plus its inverse which is negative of a we have this is equal to 0. So in case of sets for any set a union its inverse that is a complement is equal to xi that is the universal set. xi means not an empty set that is not the empty set phi and for any number a by put that by its inverse which is the reciprocal of a that is 1 upon a is equal to 1 and in case of the sets we have a intersection its complement that is a complement would be equal to the empty set phi which is not the universal set xi. For sets we have the item important laws according to which we have a intersection a is equal to a and a union a is equal to a but in comparison to this in arithmetic we have x plus x is equal to 2x and not also x into x is equal to x square and not x. So here we find there is an algebra of sets which has some axioms in common with the ordinary algebra but it has also many differences. Next we discuss mathematical system of sets we know that for any two sets there is a unique union as well as a unique intersection. So therefore we can say that taking the intersection the union the property of being a binary operation. The complement with respect to universal set is unique and so therefore we can say that the complementation is a single ugly operation and therefore it is possible to construct a mathematical system of sets in which the sets are the elements. And also any class of sets which is closed under the three operations union intersection complementation constitute system of sets. In this presentation we have understood the algebra of sets and the mathematical system of sets.