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Part 2: For questions in this section, please show any work (where necessary) _14. (10 points) For any sets A, B, and Cwhere AgB and BEC , show that AgC . Use a nar...

Question

Part 2: For questions in this section, please show any work (where necessary) _14. (10 points) For any sets A, B, and Cwhere AgB and BEC , show that AgC . Use a narrative proof.15. (10 points) Devise an algorithm that finds the product of all integers in a list & 3n a, where n22_

Part 2: For questions in this section, please show any work (where necessary) _ 14. (10 points) For any sets A, B, and Cwhere AgB and BEC , show that AgC . Use a narrative proof. 15. (10 points) Devise an algorithm that finds the product of all integers in a list & 3n a, where n22_



Answers

In Problems $21-26,$ use mathematical induction to prove that each $P_{n}$ holds for all positive integers $n .$ $$ P_{n} \text { in Problem } 14 $$

So we need to approve the following theorem that this some even called you too. And And then plus one. Good in the induction. So in induction pr one Just going to be four and that in fact is difficult to Um two times and she has the time is wrong, multiplied by one plus one. Which is to that is correct. And now we need to prove that if PRP is correct Many of that is equal to two K mm. Kay plus one then in fact The Okay. Lost one is also correct. Which means did you go to two K plus one? I'm K plus two. Now what is P of K plus one? P F K plus one? He just equal to PFK plus two because you killed PRK excuse me. Class four times K plus one. Right? That's what you had from one step to the next. And um well what is PRK PFK is two K. I escaped us one. If he adds four times K-plus one that just kind of give us two times a plus tube Times K-plus one. Which is in fact what do you need to prove? So you prove the base case. You proved one step to the next. And that process for all values

So this question we need to prove the theory mp event that if it's equal to eight to the power of five times a to the power event. And that's equal to a to the power of N Plus five. So Especially serving p. 0. 1 that if that is a five times A. 01, then that is going to be called to that. So what we have is, well, I cannot really prove this rule that easily. Well, technically what I have is you just add the song you get A. 06. I mean that's like Ruben, we know that for sure. And um we can't do that for P. 02. Also is going to give us a F five times a day or two. Going to give us this home mail seven and so on and so forth like this. So what we need to prove is if you go all the way to here then Okay, well that's gonna be a of why I'm later okay, going to be equal to it to the power of cape last five. Right? And we had if this is true then What is PFK- Plus 1? They were saying based on that theorem, that should also be okay plus six. That's what we need to get. So let's divide the two, you know, JFK plus one is equal to a to the power of five times eight to the power of K plus one. And that divided by P. F. K. It's going to be equal to let me just erase this april sign, It's going to be a call to that number right there over eight or 5/8 to decay. He's going to give us easily a, therefore, in order to go from p f k t p r cape, last time you need to multiply by a, which means you have K plus one is equal to eight times PRK Which is eight times Hey, to the five times a day today. Okay. Which will be A to the K plus six. So we have proven.

So we need to prove this theory with an induction. And let's serve with pr one. Pr one will state at eight to the power of five to the power of one. We just tickled to eight to the power of five times one or five. Is this true? Well what is this? This is a to the power of five multiplied by itself. Including itself. Once many justice. So the first case is easy to prove to prove. Now let's Kpfk is true. Then A. F. K. To the power L. A. To power A. And time to the power of K. To people to A to the power of five. Okay. And we say let's just assume that this is true. Using that we need to prove that peel a plus one is the call to a of five times cave plus one or just five K. Plus. But well, what is P F K plus juan. Really? Right. And dad is going to be a all five. All to the power of K plus one. That's what we know it is for sure. And that is equal to a five to the power of come on to the power of Okay. Multiplied by a. of five. But empower old one had also been off of church. So that is equal to into the power of five times. P okay, because this is P F K. And what is P F K. It's a to the power of five K. Multiplied by a to the power of five That I did. Everything together gives us 8 to the power of five. Okay. Plus five. Which is what we needed to prove here. So we have proved the base case. We have proved. Where we need to go from one step to the next and the industry, we can improve our solution for all real.

We're going to prove that this is a hologram by using the M five and now to do that, we first see that a B C. D. Is a parallelogram. That means that a B is parallel to D. C. And therefore a m must also be parallel and see. So that's what that out I am is Ella and see? And now we also see that if N is the midpoint of DNC, that means that we must have this length in this length are equal to each other. But then we also see the M is the midpoint of a and B, and so am I must be the same length is and B but also since we had given them a B C. D s a parallelogram, uh, that means a bee has to have the same length. It's D c. And so all of these side links that we just labeled also to be equal to each other. So that means that a M has to also be congruent to end, See, And since I am an NC of both can grant in parallel by being on 55 this is a pair of little ingrate


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