 Hello and welcome to the session. In this session we discussed the following question that says let D14 be a set containing the elements 1, 2, 7 and 14 define the operation of sum, product and complement in D14 by A plus B is equal to LCM of A and B that is the least common multiple of A and B. A into B is the GCD of A and B that is the greatest common divisor of A and B and A complement is equal to 14 upon A where A and B are the elements of set D14. Show that the algebraic system consisting of the set D14 operations of sum, product, complement 1 and 14 is a Boolean algebra. Consider the algebraic system consisting of the set B operations of sum, product and complement. Now this would be a Boolean algebra if the following laws satisfied. The first law is the closure law. Next law is the commutative law. Then we have the distributive law. Then next associative law. Then next is the identity law. And then we have the inverse law. All these laws are satisfied. Then the given algebraic system would be a Boolean algebra. This is the key idea to be used in this question. Let's now proceed with the solution. We have given a set D14 with elements 1, 2, 7 and 14. Now for A B belonging to the set D14 the operation of sum is defined as A plus B is the LCM of A and B. Then A into B is the GCB of A and B. Also A complement is equal to 14 upon A. Now we are supposed to show that the algebraic system consisting of the set D14 the operations of sum, product, complement and 1 and 14 is a Boolean algebra. So for we will construct the tables corresponding to the operations of sum, product and complement. Now the table 1 defines the operation of the sum as A plus B is the LCM of A and B. So 1 plus 1 would be the LCM of 1 and 1 which is 1. LCM of 2 and 1 would be 2. LCM of 7 and 1 would be 7. LCM of 14 and 1 would be 14. In the same way we will fill the rest of the table. So this is the table defining the operation of the sum. Next we have the second table which defines the operation of the product. Now A into B is the GCB of A and B. So 1 into 1 would be GCB of 1 and 1 which is 1. GCB of 2 and 1 is 1. GCB of 7 and 1 is 1. GCB of 3 and 1 is 1. So in this way we will fill the rest of the table. So this is the table defining the product operation. In the same way we have the third table defining the operation of the complement. Now as here we have A complement is equal to 14 upon A. So if A is 1 then A complement for 1 would be 14 upon 1 which is 14. Then if A is 2, 14 upon 2 is 7. So complement of 2 would be 7. Now A is 7. So 14 upon 7 is 2. So complement of 7 is 2. Now if A is 14, then 14 upon 14 is 1. So complement of 14 is 1. Now we will see if all the laws are satisfied. First consider the closure law. Consider elements 1 and 2 which belong to the set D14. Now 1 plus 2 is equal to from the table we have this is equal to 2. So this 2 also belongs to D14. Now consider 1 into 2 from the second table we have 1 into 2 is 1 which belongs to the set D14. Therefore from this we conclude that if 1 and 2 are the elements of set D and 14 when 1 plus 2 also belongs to D14 and 1 into 2 also belongs to D14. So we can say that if A and B are the elements of set D14 then this means that A plus B also belongs to D14 and A into B also belongs to D14. Thus we can now say that D14 is the closure law is satisfied. Now consider the commutative law also we consider the elements 1 and 2 of the set D14. Now from the table 1 we have 1 plus 2 is equal to 2 also 2 plus 1 is equal to 2 so this means that 1 plus 2 is equal to 2 plus 1. Now consider 1 into 2 from the table 2 which is equal to 1 then 2 into 1 from the table 2 is equal to 1. So you conclude that 1 into 2 is equal to 2 into 1. Therefore that if A and B are the elements of the set D14 then A plus B is equal to B plus A A into B is equal to B into A. This is the commutative law. Thus the commutative law is for the next law that we consider the commutative law for this we consider the elements 1, 2 and 7 belonging to set D14. Now consider 1 plus 2 plus 2 plus 2 plus 2 plus 2 plus 2 plus 2 plus 2 plus 2 plus set D14. Now consider 1 plus 2 plus 7 the whole now from table 1 we have 2 plus 7 is 14 so here we have 1 plus 14 now 1 plus 14 from table 1 is 14. Now consider 1 plus 2 the whole plus 7 is equal to 1 plus 2 is 2 from table 1 or 2 plus 7 from table 1 is 14. Therefore we conclude that 1 plus 2 plus 7 the whole is equal to 1 plus 2 the whole plus 7. Now consider 1 into 2 into 7 the whole is equal to 1 into now 2 into 7 is 1 from table 2 now 1 into 1 is 1. Now consider 1 into 2 the whole into 7 is equal to 1 into 2 is 1 into 7 now from table 2 1 into 7 is 1. So this means that 1 into 2 into 7 the whole is equal to 1 into 2 the whole into 7 and so we can say that if A, B and C are 3 elements belonging to set D14 then A plus B plus C the whole is equal to A plus B the whole plus C and A into B into C the whole is equal to A into B the volume to C. Thus we can say that the associative law is satisfied. Consider the distributive law we consider 1, 2 and 7 the elements of the set D14. Let us first check the distributivity of the product operation over the sum operation. So for this we consider 1 into 2 plus 7 the whole now this would be equal to 1 into now from table 1 2 plus 7 is 14 now 1 into 14 from table 2 is 1 so this is equal to 1. Now consider 1 into 2 the whole plus 1 into 7 the whole from table 2 1 into 2 is 1 plus 1 into 7 is 1 and from table 1 1 plus 1 is 1 so they both are equal that is we have 1 into 2 plus 7 the whole is equal to 1 into 2 the whole plus 1 into 7 the whole. So this shows the distributivity of product over the sum. Now consider 1 plus 2 into 7 the whole this is equal to 1 plus now 2 into 7 from table 2 is 1 now from table 1 1 plus 1 is 1. Now consider 1 plus 2 the whole into 1 plus 7 the whole. Now from table 1 1 plus 2 is 2 into 1 plus 7 the whole. Now from table 2 2 into 7 is 1 so they both are equal that is 1 plus 2 into 7 the whole is equal to 1 plus 2 the whole into 1 plus 7 the whole. So we can now say that if A B and C are 3 elements of set D 14 then A into B plus C the whole is equal to A into B plus A into C this shows the distributivity of the product over the sum then A plus B into C the whole is equal to A plus B the whole into A plus C the whole. This shows the distributivity of the sum over the product and thus we can now say that the distributive law X we consider the identity law meant A which belongs to the set D 14. Now from table 1 we observe that A plus 1 is equal to A which is same as 1 plus A. If we consider A as 1 then 1 plus 1 is 1 if A is 2 then 2 plus 1 is 2 7 plus 1 is 7 14 plus 1 is 14 therefore we can say that this 1 is the D element for the sum operation. Now from the table 2 we observe that A into 14 is equal to A which is same as 14 into A. Like if A is 1 then 1 into 14 is 1 2 into 14 is 2 7 into 14 is 7 14 into 14 is 14. So therefore from this we can say that A plus B is equal to A that this 14 is the identity element for the operation product. Next law that we have is inverse law consider A to be equal to say 1 then from the table 3 we have A complement is 14 that is 1 complement is 14. Now 1 plus 14 from the table 1 is 14 also 14 plus 1 is 14. So therefore we have 1 plus 14 is equal to 14 which is same as 14 plus 1. So we can say for any A belonging to the set D 14 A plus A complement is equal to 14 which is same as A complement plus A and this 14 is the identity element for the operation of the product. Now consider A into A complement that is 1 into 14 from table 2 1 into 14 is 1. Now A complement into A is 14 into 1 which from the table 2 is 1. We have A into A complement is equal to 1 is equal to A complement into A for any A belonging to the set D 14 and this 1 is the identity element for the operation of the sum. So this shows that for an element A belonging to the set D 14 there exists its inverse A complement such that A plus A complement is equal to 14 is equal to A complement plus A where this 14 is the identity element for the operation of product and A into A complement is equal to 1 which is same as A complement into A where this 1 is the identity element for the operation of sum. Hence all the laws are satisfied. So we can say the algebraic system consisting the set D 14 operations plus product complement 1 and 14 is a Boolean algebra. So this completes the session. Hope you have understood the solution of this question.