 Hi, and how are you all today? Let us proceed on with the question that's given to us. It says, let A and B be sets. If A intersection X is equal to B intersection X is equal to 5 and A union X is equal to B union X for some set X, show that A is equal to B. So let us proceed on with the solution and let us make use of the things that are given to us. It's given that A intersection X is equal to B intersection X is equal to 5 and A union X is equal to B union X for some set X. Let's start with our proof. Now, we can write A, set A as A union 5, right? We can write it down like this. Now, we can say A is equal to A union. Now in place of 5, we can write B intersection X because it is given that 5 is equal to B intersection X. It is given to us in the question. We are just making use of it. B intersection X, proceeding on. We can have using distributive law. We can say that A union B, A is equal to A union B intersection A union X further. Now in place of A union X, we can use B union X that is given to us in the question that A union X is equal to B. A in this, that is given to us that A union X is equal to B union X. So in place of A union X, we can write B union X. Again, by using distributive law we can write it as B union A intersection X. And as we know that A intersection X is given to us in the question as 5, we can substitute the value of B union. A intersection X now can be written as 5, which will give us that A is equal to B. So this is what we were supposed to prove and hence we have shown the required thing. So I hope you enjoyed the session. Do remember all the properties and laws that you started before proceeding on with the solution. Bye for now.