 Hello and welcome to the session. In this session, let us define Boolean Algebra. Boolean Algebra is basically mathematical systems based on the properties of sets. So we can say a Boolean Algebra is any mathematical system which consists of a set of elements closed under two binary operations intersection and union and satisfying the following conditions or axioms. The first one which says that both the intersection and the union are commutative then next one says that both the intersection and the union are associative then next we have that each operation is distributive over the other like we say for any three elements a, b and c a intersection b union cd whole is equal to a intersection bd whole union a intersection cd whole. So this means that intersection is distributive over the union then next we have a union b intersection cd whole is equal to a union bd whole intersection a union cd whole. So in this case union is distributive over the intersection. Next we have the identity element for both the intersection and the union. First we have for intersection the identity element may be named as 1 then for union the identity element may be named as 0. Next we have for any element a there exists a complement such that a intersection a complement is equal to 0 which is the identity element for the union and a union a complement is equal to 1 which is the identity element for the intersection. Let us now discuss the general definition for Boolean algebra of any number of elements. A non-empty set b with two operations of sum and product that is we can also say that an algebraic system consisting of a set b of elements a, b, c and so on and two binary operations called the sum which is denoted by a plus sign and the product denoted by this dot is called a Boolean algebra if and only if for all a, b, c belonging to the set b the following axioms closure property according to which we have that the set b is closed under the three operations so we say for a, b belonging to the set b we have a plus b also belongs to the set b a into b also belongs to the set b and a complement belongs to set b also b complement belongs to set b next we have the commutative property that is for a, b belonging to the set b we have a plus b is equal to b plus a also a into b is equal to b into a so this shows that the operations of sum and product are commutative next we have the associative property consider a, b, c belonging to the set b so a plus b plus c the whole is equal to a plus b the whole plus c this shows that the operation of sum is associative then we have a into b into c the whole is equal to a into b the whole into c this shows that the operation of the product is associative now the next property is the distributive property each operation is distributive over the other that is for a, b, c belonging to the set b we have a into b plus c the whole is equal to a into b the whole plus a into c the whole this shows that the product is distributive over the sum then we have a plus b into c the whole is equal to a plus b the whole into a plus c the whole this shows that the sum is distributive over the product next we discuss about the identity element for all a belonging to the set b we have a plus 0 is equal to 0 plus a and this is equal to a here this 0 is the identity element for the operation of the sum also we have a into 1 is equal to 1 into a is equal to a and here this element 1 is the identity element for the operation of the product and the 0 and 1 belong to the set b next we discuss about the inverse element for each element a belonging to the set b there exists a complement which belongs to the set b and this a complement is the inverse of the element a and we have a plus a complement is equal to a complement plus a which is equal to 1 also a into a complement is equal to a complement into a is equal to 0 so for each element a belonging to the set b there exists the inverse of the element a which is a complement that belongs to the set b such that a plus a complement is equal to a complement plus a which is equal to 1 and this one is the identity element for the operation of the product then we have a into a complement is equal to a complement into a is equal to 0 which is the identity element for the operation of the sum as we can observe that in this general definition of the Boolean algebra the union is replaced by the sum and the intersection is replaced by the product and we have replaced the sets by the numbers a b and c in Boolean algebra we have a plus a complement is equal to 1 this one is the identity element for the operation of the product and in sets here we have a union a complement is equal to thy which is the universal set also a into a complement is equal to 0 which is the identity element for the operation of the sum and in sets here we have a intersection a complement is equal to 5 which is the null set here this thy as the multiplicative identity as we have a intersection thy is equal to a and this 5 acts as the additive identity as we have a union 5 is equal to a for all a being the subset of the universal set thy next we discuss duality in Boolean algebra in a Boolean algebra the dual of any statement see statement obtained by interchanging the signs of plus that is the sum and the product then 0 and 1 in the original statement and we also know that complementation is a self dual this holds true from the principle of duality which states that the dual of any theorem in a Boolean algebra is also a theorem then in any law or theorem on sets union and intersection are interchanged and also the null set phi and the universal set xi are interchanged the new law or theorem which is valid by the principle duality likewise in any law or theorem on logic if this junction and conjunction are interchanged false and true are interchanged then the new theorem or you can say the law is valid by the principle of duality like if you consider the statement 0 plus x the whole into 1 plus y the whole is equal to x then the dual of this statement would be obtained by interchanging the plus and the dot 0 and 1 so its dual would be obtained as 1 into x the whole plus 0 into y the whole is equal to x so this is how we can find the dual of any statement this one please see session hope you have understood the definition of Boolean algebra and also the duality in Boolean algebra