 Hi and welcome to the session. I'm Priyanka and let us proceed on with the question which is given to us It says using properties of sets show that the first part is a union Bracket a intersection b is equal to a and then we need to show that a Intersection a union b is equal to a Now before proceeding on with the solution we should be well versed with the vane diagram that will help us in solving it Let this be the universal set. We're talking about and this be Set a And this be set b right now the portion Which is common to both of them? Represents a intersection b, right? So the knowledge of this whole vane diagram is the key idea that we're going to use In order to proceed on with our solution Let us proceed on Then the first part we need to show that a union a Intersection b is equal to a right now here we are talking about a Union with a intersection b a intersection b represents the green part over here and if we talk about this is Set a and if we subtract b from it it will be a Set which will be represented by this color it means Set a excluding each and every element of set b that is also common to the b part and This represents b minus a That includes only those elements which are present in a and not in b right now if we have the set a This whole set and we combine the Intersection with it what will be the set that we'll obtain it will be this whole Set a isn't it so by using the Properties we'll have Now we know that set a if we are talking about it from the universal set. It is a intersection b Union the universal set That is We have used the property using Any set intersection with the union set will give us the union set itself and Any set having an intersection with the union set will give us the set itself, right? Now proceeding on these are the property that you must know Here using the distributive law we can say that a intersection b Union a intersection the universal set proceeding on We can write down a intersection b here Union now a Intersection union will give us which set as we have discussed above any set Intersecting with the union set will give the set itself so here we can write the name of the set itself using That x intersection the union set will give us the set itself, right now using commutative law we can say that a union a intersection b has resulted into Set a We forgot writing a Right, so this was what we was supposed to prove and hence we can write down that we have proved it This is a proof that we have Done using all the laws, but if we use the vain diagram it also suggests that when a is Having a union with a intersection b it will result into set a only, right? Let's proceed on with the second part now here. We need to show that a Intersection a union b is equal to a now one of the laws that we have studied that says as x intersection phi gives us phi only and x union phi will give us the set itself we can write down that a is equal to a union b intersection Phi can't you write because Here a b intersection phi means phi only and a union phi will give us A itself so we can write it in this order set a can be written in this order also now using the distributive law We can say that a is equal to a union b Intersection a union phi now what will a union phi will result into yes You're absolutely right will result into the set itself So we have a union b intersection a equals to a now using commutative law We can say that a is equal to a Intersection a union b right this was what we were supposed to prove And this completes the entire question that was given to us So I hope you enjoyed the session do remember the laws and the properties that you have started Before proceeding on with all the questions. Bye for now