 Hi, and how are you all today? Let us do the following question together. It says if the universal set u is equal to elements 1 2 3 4 5 6 7 8 9 a is having elements 2 4 6 8 and b are having elements 2 3 5 7 verify that for the first part a union b Complement is equal to a complement intersection b complement and for the second part a Intersection b's complement is equal to a complement union b complement. So we need to verify Them so let us start with our solution of The first part now here. We need to verify a union b is complement is equal to a complement intersection b complement Now, let us write what we have the universal set elements Then we have the elements of set a which are two four six eight The elements of set b that are two three five seven So first of all let us find out a complement as we know a complement will be having elements which are present in the universal set but Which are not which do not belong to set a so they are one three five seven nine right similarly we'll find b complement that will be one four six eight and nine Right so for the left-hand side Let us find out a union b Complement that will be First of all We'll find a union b that will be One That will be Writing down the elements of set a with set b The answer will be two three four five six seven eight So a union b complement will be what the elements which are present in the universal set But are not a part of a union b that is what one and nine only Similarly, we'll find out the RHS a Complement intersection b complement write down a complement one three five seven nine Intersection b complement one four six eight Nine so the answer will be the elements which are common to both of them are One and nine So from first and second we have LHS equals RHS and we can write hence Verified so this completes the first part Now the second part says verify a intersection b complement Is equal to a complement union b complement? So let us first find out left-hand side a intersection b so which are the terms which are common to both a and B that is only one element that is two so a Intersection b complement will be having all the elements other than two that will be one three four five Six seven eight and nine Fine, let this be the first part Now right inside we need to find a Complement union b complement So it will be first of all a complement are having one three five seven nine as its element union one four Six eight and nine so what will be their result of their union and that will be one three four five six seven eight nine, let this be the second part now from One and two We have LHS equals RHS and we can write down hence Verified so like we did in our junior classes first we'll find out for left-hand side Then we'll find out for right-hand side then we'll compare the both sides and then we can say whether they are equal or not So I hope you enjoyed the session bye for now