Hi. In this question, we are asked to show that the said, uh equation here is true for any positive integer in on. We asked to use induction even though I believe proving it directly from set properties. Much simpler, but it's an exercise. So let's do it. You see induction less this that maybe p of in first step the basic step. We would have just a one intersect be on both sides, right? And they are obviously equal now for in that keeps that suppose that it is true for P l n For some, in I Wish order, we will show that is true fall in plus one as well. And here you have many ways to do it. You can try to use the information from P O in to prove P up in plus one. Or you can also use some information about said that we know is true to prove it. And it was still like the proof by induction will still be well, all right, I'll do shoes the latter. So let's go over the statement off P and pass one on how you prove it. So first, since I'm proving in the equation. All set, I would I will prove that both of them being a subset off each other, Right? First we have this. Ah, this direction. So suppose x ys and element in this set on the left hand side, it would mean since this is intersection, it would mean that X is in some a J. We don't know what it is. Doesn't matter. And it is also inside b so inside a J and B therefore, it is in the intersection between this too, right? By just the definition of intersection, that means it must be inside this June. The end off off a I the sick be because a J in the second is in there somewhere. All right, It's nothing. Complicate that. So we have a show. One implication specifically Is this one The union is that B? He's a something off union off intersection, right? And we can do the same fall for the reverse implication. So it will start like let X be in this union off intersection sins they are joined by by union. The meaning of this is that eggs must be in some some of the terms inside here. Right? We don't know which, but it doesn't matter is in some genius iq be somewhere. And so by the meaning of this intersection, it would mean that X is inboard this a j and board. This ain't a and B all right? And so, But with with this, it would mean that X is in the union off. They won, too. Itchy. Sorry. It want to a in pass water, right? Because it is in there somewhere. So it is in this whole union thing and is in be so is inside intersection off. This would be all right. Nothing complicated there. So I have shown DeVry was like being a subset. We started the right hand side. Now we arrive at the left hand side. When I show both board us said is a subset off each other. It would mean the state may everyone. So it means that where should I write this? It means the union. Oh, a 12 a and plus one in the sick Me is equal to a one in the state bee You in the n onto impossible on music Be all right that Yes, nothing public it going on it Just symbol said Properties. And so we have shown that the statement is true for peon plus one and so back to our induction. We just showing the inductive step is true. So this together with basic step, means that this statement is true. Fall all positive integer in buy meth induction and that is it. Thank you.