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Determine which functions below define surjections from N X N to N Give a one-line explanation_ f(x,y) x +y f(x,y) = xy...

Question

Determine which functions below define surjections from N X N to N Give a one-line explanation_ f(x,y) x +y f(x,y) = xy

Determine which functions below define surjections from N X N to N Give a one-line explanation_ f(x,y) x +y f(x,y) = xy



Answers

Determine whether each of these functions from $Z$ to $Z$ is one-to-one.
$$\begin{array}{ll}{\text { a) } f(n)=n-1} & {\text { b) } f(n)=n^{2}+1} \\ {\text { c) } f(n)=n^{3}} & {\text { d) } f(n)=[n / 2]}\end{array}$$

All right. So here we have two different functions to investigate. The first is to see whether the function of end being equal to two to the end plus one is big O of two to the power of end. Figure that out. And so we're just gonna let initially k equal zero. Okay, so that implies that end must be greater than zero. It's the lowest, uh, down there you could choose that appropriate assumption, and then we'll take a look. And we see that the absolute value death event that is equal to the absolute value of two to the end. Plus one. Okay, this is equal to two times two to the end, which is equal to two. Has as a value of two to the power, then. Okay, So if we let c equal to and by big o notation the absolute value of the van, just gonna be equal to abso dahlia two times and plus one must be less than or equal to two times. That's the value of two to the end, right? We already see that. That is true. Okay, So in this case, ever then being able to to to power them plus one is big O of two to the end and the other function to investigate is effort then being equal to two to the power of to end right. We still have the rg of n is equal to two to the here. So this time we're gonna assume that their big over it's gonna assume that two to the power of to end is big O of two to the power. And well, if this were true, then it must be the case that absolute value two to the power of two n would be less than or equal to some constant times M flew value of two to the power of end for some and being greater than okay, yes. Let's assume that sea is greater than or equal to zero. Okay, it's a safe assumption. Absolute values. They're always not negative. Do we get that too? To the end times T n is equal to two to the power of to end which is equal to the absolute value of two to the power to end, which must be lesson or equal to see it's the absolute value two and is equal to see times two to the end. Nice. We let Kate be greater than or equal zero. She's a safe, low boundary, since to the end times to the N is less than equal to see times two D and weaken Divide by two To the power of Enright is that leaves us with two to the power of n is less than or equal to see. But we have a contradiction, right, because when you solve for and you get that and it's equal to log to you of C plus one, then we plugged it in. You end up with is that two to the N is equal to to you to the power of log to see, plus one which is equal to seed plus one. But if two to the end is less than equal to see than this implies, that C plus one must be less than or equal to see right, and that's clearly wrong. Okay, so that's never true. And so therefore uh, f of X. Being equal to two to the power of to end is not and the big O of two to the power, then

I e one equal to chew. And then we have a general term endless one Nico to the endless one times a nd value. But you and for the end, greater it could you one here we can try to find a ate you equal itude two times their previous term too. Even made you so get equal to the chew 83 equal to the three times their previous term Defending bite. You gonna go to three a far ico June The farm attempts their previous time development you and we get equal. Jude Ah ah six a five echo to five times debris was term depending but you. Then we get equal to the fit deal and so on. Here. Notice that we can write the A one in We go to the one factorial on then and dividing by Ah ah half ah half We can register into the to power off the ah one minus two and then similarly this one equal to the two factorial dividing by two power off the two wants to hear we have when we go to the three. Fraternal If I didn't mind the to about three minus one and they will hand off off a Toyota dividing by Jew. About four minutes. One if I'm honest you here. Sorry. And this one recorded a five fraternal development. You power bi minus two. So, in general, held any cogent and Victorian, if any, May 2 hour and minus two.

Hello. We're to find the function F X That did you cause to and function of NX at two X. Okay, so the function and that's where it is going There. Three x squared plus X squared three X y plus X square minus of ice cream plus one So we will find the value functional at ext works. So it will because 23 x two X plus X squared minus off two x will describe plus one. Did you pursue six X squared plus X squared minus of four. Access card. Last one. So it will because to three x squared. Yeah, plus one. So the value of FX will because 23 X squared plus one. Okay. And this is the answer. I hope you're not sure. Thank you.

Ah So we have a function that takes um two words in the english had weapons and gives us a new world. So we want to see if it's in my objective. So we'll see if it's injected for us. So we'll be two different inputs and see if we get yes. Oh that's it's one equal to nuts and why one so what and X. Two equal to and why to equal to observe that this two inputs are not the same. And we're sure if its objective we should have the outputs also. But that's all that's one Why one when we combine them we get the world uh right And this is the same thing we get when we combined next to right to. So this is not injected so not collective. So it can be a project.


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