 Hello and welcome to the session. In this session we discuss Theorums of Boolean Algebra. First of all we consider a Boolean Algebra consisting of a set B and the binary operations of sum and product. Now consider x and y to be elements of the set B. Now let's proceed with the Theorums. First we have the item potent law. In this we have x plus x is equal to x and x into x is equal to x. Where this x is the element of the set B. Let us now see the proof of this. First of all we will prove x plus x is equal to x. Now consider x it could be written as x plus 0 by the property of the identity. Since by the property of identity we have a plus 0 is same as 0 plus a and this is equal to a. So we can write x as x plus 0. Now next by the property of inverse we have a into a complement is 0 which is same as a complement into a. So further we can say x is equal to x plus. Now in place of 0 we can write x into x complement. This is by the property of inverse. Now further we will use the distributivity of the sum over the product that is we would use this distributive law to the right hand side and so we get x is equal to x plus x the whole into x plus x complement the whole. This is by the distributive law. Now again we consider the property of inverse by which we have a plus a complement is 1. So further we can say x is equal to x plus x the whole into 1 that is x plus x complement is 1. This is by the property of inverse. Now again by the property of identity we have a into 1 is equal to 1 into a is equal to a. So using this we have x is equal to x plus x that is x plus x the whole into 1 is x plus x. This is by the property of identity and therefore we now have x plus x is equal to x and this is what we were supposed to prove. Now consider the second part in which we need to prove x into x is equal to x. For this again we consider x which could be written as x into 1 by using the property of identity that is this property. Further by the property of inverse we can write this 1 as x plus x complement. So here we have x equal to x into x plus x complement the whole. This is using the property of inverse. Now using the distributivity of the product over the sum that is using this distributive law we further have x equal to x into x the whole plus x into x complement the whole. This is using the distributive law. Now further we have x is equal to x into x plus x into x complement would be 0 using this property of inverse. Now by the property of the identity we have x is equal to x into x that is x into x plus 0 would be x into x. So therefore we now have x into x is equal to x and this is what we were supposed to prove. Now x into x equal to x is the dual of x plus x equal to x. It means that in this statement if we replace this plus by the dot then we get x into x is equal to x and so this is the dual of x plus x equal to x. And each step of the proof of the second part is the dual of the first part. So hence proved the idempotent law. Now let's move on to the next theorem. First part we have x plus 1 equal to 1. In the second part we have x into 0 is equal to 0. And here again x is the element of the set b. Now let us see the proof of this theorem. Consider the first part in which we have to prove that x plus 1 is equal to 1. Now by the property of inverse we have a plus a complement is 1 which is same as a complement plus a. So we can say that 1 is equal to x plus x complement. This is by inverse. Now by using this property of identity we can say that x complement is same as x complement into 1. So here we have 1 is equal to x plus x complement into 1. This is by the identity. Now further we will use the distributivity of the sum over the product that is this law. So we have 1 is equal to x plus x complement the whole into x plus 1 the whole. Further we can use the property of inverse and so we have 1 is equal to x plus x complement could be written as 1 into x plus 1 the whole. So this is by the property of inverse. Now further by the property of identity we have 1 is equal to 1 into x plus 1 the whole is x plus 1. So therefore we now have x plus 1 is equal to 1 and this is what we were supposed to prove. Now in the second part we need to prove x into 0 is equal to 0. Now x into 0 equal to 0 follows from duality that is in place of plus we put dot in place of 1 we put 0. So we get x into 0 is equal to 0. So hence proved the second theorem. Let's now proceed with the third theorem which is the law of absorption. In this we have x plus x into y is equal to x this is the first part and in the second part we have x into x plus y the whole is equal to x where x and y are the elements of the set B. Let's see the proof of this now. First of all we will prove x plus x into y is equal to x. For this consider x plus x into y now using this property of identity we can write x as x into 1 plus x into y. Now next we would use this distributive law showing the distributivity of the product over the sum that is this law. So this is equal to x into 1 plus y the whole this is by the distributive law. Now further we have x into y plus 1 the whole this is using the commutative law as we know that the sum is commutative. So 1 plus y is y plus 1. Now from the theorem 2 part 1 we have x plus 1 is equal to 1 so this is equal to x into 1 that is in place of y plus 1 we have written 1 this is using theorem 2 part 1. Now further by identity we have x into 1 is x we now have x plus x into y is equal to x and this is what we were supposed to prove. So we have proved this here. Now in the second part we need to prove x into x plus y the whole is equal to x and this result follows from duality. So hence we have proved this third theorem also. Now in the next theorem we have this complement the whole complement is equal to x. Let's see we prove for this now. Now consider x which belongs to the set B. Now by this property of identity we can write this x as x plus 0. Now as we know a into a complement is equal to 0 a complement the whole complement would be the inverse of a complement so in that case a complement into a complement the whole complement would be equal to 0. So further we have x is equal to x plus in place of 0 we write x complement into x complement the whole complement. This is by the property of inverse. Now by using this distributive law we further have x equal to x plus x complement the whole into x plus x complement the whole complement the whole. This is using the distributive law. Now further from the property of inverse we have a plus a complement is 1. So here we have x equal to x plus x complement which would be 1 into x plus x complement the whole complement the whole. This is by the property of inverse. Now as a complement the whole complement is the inverse of a complement so we can also say that a complement plus a complement the whole complement is equal to 1. So further here in place of 1 we can write x complement plus x complement the whole complement this whole into x plus x complement the whole complement the whole. This is by the property of inverse. Now further we can write this as x equal to x complement into x plus x complement into x complement the whole complement plus x complement the whole complement into x plus x complement the whole complement into x complement the whole complement. Now from the property of inverse we have a into a complement is 0. So further x is equal to x complement into x which is 0 plus x complement into x complement the whole complement would would also be 0 plus x complement the whole complement into x plus now from the idempotent block we have x into x is equal to x so here x complement the whole complement into x complement the whole complement could be written as x complement the whole complement this is by the theorem 1 part 2 and also by the property of inverse we have this 0 and this 0 so we now have x equal to x complement the whole complement into x plus x complement the whole complement now further we have x equal to x complement the whole complement into x plus now this could be written as x complement the whole complement into 1 this is by the property of identity now further we will use this distributive law so we have x equal to x complement the whole complement into x plus 1 the whole this is using the distributive law now from theorem 2 part 1 we have x plus 1 is equal to 1 so here we have x is equal to x complement the whole complement into 1 this is by the theorem 2 part 1 now further we have x equal to x complement the whole complement this is by the property of identity so hence we have x complement the whole complement is equal to x and so we have proof this theorem now we move on to the next theorem in which we have one complement is equal to 0 and 0 complement is equal to 1 now let's see the proof of this first of all let us prove one complement is equal to 0 for this consider one complement which could be written as one complement into 1 this is by the property of identity now using this property of inverse we have one complement equal to one complement into 1 which is equal to 0 so we have one complement is equal to 0 this is what we were supposed to prove now in the second part we have to prove that 0 complement is equal to 1 and this follows that is 0 complement equal to 1 follows by duality that is in this result we replace 1 by 0 and 0 by 1 and so we get 0 complement is equal to 1 hence proved this theorem so this completes the session hope you understood the theorems of Boolean algebra