1. Prove that if nis an odd integer, then[n+H+pn-H =2n.Specify the type of proof you are using:Your proof: In my proof I use...

Prove that these four statements about the integer $n$ are equivalent: $(i) n^{2}$ is odd, $(i i) 1-n$ is even, $(i i i) n^{3}$ is odd (iv) $n^{2}+1$ is even.

Okay, So in this question, we want to prove that the floor and divided by two is equal to end over two for n even an end once one divided by two went in on. So how do we do this? It was first it up. So that and be even. Well, if it is, even then we know that exists and interject and m such that and it's equal to chew em on. That is m is equal to and why. So we substitute this into our floor function. So to em, divided by two the truth is capital out. So this is for him since M isn't it? Georgia? The floor damages. We also know that M is equal to end it. What about you? So the first part sounds fine dispute. Now we let and be on. So in this case, that exists an easy job. Em such dense. And is it to shoot and plus one? Well, this is me. It means that I am so we subtract the one over. And you did too. The end If it was end once. One black budget. Now you put this into our floor function. But this is going to be floor all to M plus one divided by two. So this is in just a, huh? Now, how do we solve this? So am I, Is an insurgent and m plus one. And hop is right here. So because we're taking floor, we're going to take the incident that is right next to us. So this is just beat him. But when I missed the end, while it's one divided by two and minus one people about to so therefore the second part satisfied with trivial they're combining these two things together. We have our question.

Okay. So here um in part a we want to prove that if n is odd then and squared is odd. Well if n is odd then we know that there exists some integer K. Such that N is equal to two K plus one, Right? Because the number is even if it's equal to two K. For some integer K. And if it's odd, well that's one more than an even number. Therefore it's going to be equal to two K plus one. Okay? Um So if you just plug this in for and squared, what do we get? So if n is odd and is equal to K plus one, therefore and squared is going to be equal to two K plus one squared, which is going to be equal to y two K plus one times two K plus one gives us a four K squared plus four K plus one. But do we can actually factor this as the four k squared plus four K. Just factors as two times k squared plus two K. Right. That's too that's um you know, two times two k square. That's gonna be a four K squared plus four K plus one. Um But then we can just go ahead and replace because two K squared plus k, it's just some integer. Just call that some integer. Maybe some integer. I don't know jay. So therefore that n squared is going to be equal to um well just two times some integer called that LBJ. So to J plus one um which shows here that ends squared is just equal to two times interview plus one. Right jay Wright Called J K if you want it doesn't matter what you just equals. En squared is equal to two times the energy plus one. Um This shows that and squared is odd. So therefore if n is odd, we then um conclude that and squared must also be odd as well. Um for part B. Okay, we want to prove this statement. Well, if and squared is odd then N it's going to be odd. Okay, well um we can prove this statement by finding and proving the contra positive, remember that a statement and its contra positive are going to either both be true or both be false. So the contra positive of this statement here um is going to be if um if n is not odd then and squared is not odd. So if n is not odd then and squared is not odd. Yeah or in other words if and if and it's not odd. That means that end is even. So if N is even then and squared is even Mhm Okay. So I have to prove this what we know that if n is even right then and is going to be equal. Um too well some um it's gonna be eager to two K Rachel and is even there exists an integer K. Such that um N is going to be equal to um Well we can write this is actually two K plus two. In other words we can always add to and multiples of two. Um to obtain another even integer. So we plug this in for and squared we then have that and squared is going to be equal to two K plus two squared. Um Which is going two be equal to um well two K plus two times two K plus two is going to be equal to four K squared plus eight K plus four. We can factor the four K squared plus eight K by factoring out um A just a two. So this could be equal to two times and you could go back to two K but only because it's the only factor out A K. We've had four K squared plus eight K is equal to two times two K squared plus four K plus four. Oh but the two K squared plus um two case clippers, four K is just another integer. So therefore therefore we have that and squared is going to be equal to two times. Some introductory call that J lo to J plus four. And we see here that um and squared will also always be even because two times some integer plus plus K plus two times some integer plus four um is always going to be even. So therefore we see that and squared um is also going to be even as well. So therefore we prove the counter positive, which proves the original statement. So I'm gonna conclude here that ends squared and squared is going to be even

We can't use the real proof to show some off to off just is. Even so before that X and Y you're home introduced. Then what we know is that my ex is being too tough for a plus one for a summer internship like, Why is being saying scene should be plus fired off? One we'll be in touch is then we add, Except what together? Because that you are numbers that X plus y y give you do a one wants to be one you choose a. That's why to be possible actu we can factor out to this is to a be possible now. A plus B plus one was just mother interview. We can walk there. This careful as QC We'll see some into jail. They can't express what he's an even number because to type my sentence